VOLUME   II 

. 

ALTERNATING  CURRENTS 

AND 

ALTERNATING  CURRENT   MACHINERY 


ALTERNATING  CURRENTS 


AND 


ALTERNATING   CURRENT  MACHINERY 


BEING  VOLUME  II  OF  THE 

TEXT- BOOK  ON  ELECTRO- MAGNETISM   AND 
THE  CONSTRUCTION  OF  DYNAMOS 


BY 

DUGALD  C.  JACKSON,  C.E. 

PROFESSOR  OF  ELECTRICAL  ENGINEER!NG  IN  THE  UNIVERSITY  OF  WISCONSIN; 

MEMBEIT  OF  THE  AMERICAN  SOCIETY  OF  MECHANICAL  ENGINEERS, 

AMERICAN  INSTITUTE  OF  ELECTRICAL  ENGINEERS,  ETC. 

AND 

JOHN   PRICE   JACKSON,   M.E. 

PROFESSOR  OF  ELECTRICAL  ENGINEERING  IN  THE  PENNSYLVANIA  STATE  COLLEGE; 
MEMBER  OF  THE  AMERICAN  INSTITUTE  OF  ELECTRICAL  ENGINEERS,  ETC. 


Nefo  gotfc 


THE   MACMILLAN    COMPANY 

LONDON :  MACMILLAN  &  CO.,  LTD. 
1896 


All  rights  reserved 


COPYRIGHT,  1896, 
BY  THE  MACMILLAN  COMPANY. 


J.  S.  Gushing  &  Co.  —  Berwick  &  Smith 
Norwood  Mass.  U.S.A. 


PREFACE. 


THE  matter  of  this  book  consists,  in  essence,  of  the  lectures 
which  have  been  delivered  for  two  or  three  years  past  by 
Professor  D.  C.  Jackson  to  the  senior  and  graduate  students 
in  Electrical  Engineering  at  the  University  of  Wisconsin,  but 
Professor  J.  P.  Jackson,  of  the  Pennsylvania  State  College,  care- 
fully revised  and  extended  the  manuscript  before  it  was  sent 
to  the  printers.  The  method  carried  out  in  the  book  is  based 
on  the  self-evident  but  little  recognized  principle,  that  methods 
which  have  proved  best  in  teaching  other  branches  of  Engi- 
neering must  be  equally  advantageous  in  treating  Electrical 
subjects.  A  treatise  on  electro-magnetism  or  on  alternating 
currents  should  therefore  deal  with  its  subject  in  much  the 
same  way  that  thermodynamics  and  the  steam  engine,  hydrau- 
lics and  hydraulic  machinery,  or  the  theory  of  structures  are 
respectively  presented  in  the  best  works  on  those  subjects. 
The  startlingly  rapid  advances  which  have  been  made  in 
our  knowledge  of  the  phenomena  relating  to  electro-magnet- 
ism and  the  electric  current,  tend  to  confuse  the  best  teachers, 
and  doubtless  account  for  the  rather  superficial  methods  used 
in  many  colleges,  in  which  results  only  have  been  presented 
to  the  student  with  little  reference  to  reasons.  This  error 
is  generally  admitted,  and  it  is  hoped  that  this  work  may, 
by  furnishing  a  satisfactory  text-book,  aid  those  teachers  who 
desire  to  improve  their  methods. 


vi  PREFACE. 

The  book  treats  of  the  fundamental  phenomena  of  alternat- 
ing currents  as  met  with  in  engineering  practice,  and  points  out 
their  controlling  principles  and  applications.  Descriptions  and 
illustrations  of  commercial  machinery  are  not  included  per  se, 
since  they  would  be  little  more  than  repetitions  of  matter  which 
is  available  in  the  current  technical  journals,  and  would  crowd 
important  material  from  the  book.  This  does  not  mean  that 
practical  data  are  excluded ;  on  the  contrary,  where  they  may 
be  useful  in  illustrating  deductions  in  the  text,  they  are  copi- 
ously used,  selections  having  been  made  for  the  purpose  from 
an  extensive  mass  of  data,  most  of  which  is  original.  A  large 
number  of  references  to  articles  are  given  in  foot-notes  for  the 
fuller  information  of  the  reader.  These  cover,  in  general,  those 
articles  which  may  be  read  in  the  original  by  the  student  with 
most  profit ;  except  in  the  chapters  on  polyphase  currents. 
Here  the  list  of  references  is  much  less  complete,  since  the 
subject  has  not  yet  been  fairly  wrought  out,  and  material  of 
overshadowing  importance  is  being  constantly  published,  so 
that  only  a  few  of  the  writings  of  most  substantial  importance 
can  be  advantageously  cited.  A  number  of  excellent  articles 
have  lately  been  published  upon  polyphase  currents  but  at  too 
late  a  date  to  be  put  into  the  plates.  Where  articles  of 
importance  have  been  published  in  a  number  of  prominent 
American  and  foreign  technical  periodicals,  the  references 
have  usually  been  made  to  the  Electrical  World  and  the  Lon- 
don Electrician. 

Throughout  the  book,  occupation  of  space  by  the  descrip- 
tion of  classical  experiments  which  have  come  to  be  of  his- 
torical interest  only,  has  been  carefully  avoided,  except  where 
it  seems  desirable  to  trace  the  natural  development  of  know- 
ledge in  a  particularly  important  subject :  as,  for  instance,  the 
subjects  relating  to  the  tracing  of  alternating-current  curves, 
methods  of  measuring  inductances,  or  practice  in  the  paral- 


PREFACE.  vii 

lei  running  of  alternators.  The  last  is  a  subject  of  much 
importance  at  the  present  time,  on  account  of  its  influence  on 
the  uniformity  of  the  service  given  from  alternating-current 
central  stations,  and  its  rapidly  increasing  adoption  either  for 
regular  operation  or  for  the  purpose  of  transferring  the  load 
from  generator  to  generator. 

Much  very  important  matter  which  heretofore  has  been  found 
only  in  technical  periodicals,  and  sometimes  has  been  unavail- 
able for  use,  is  to  be  found  in  this  book.  In  a  number  of  cases 
original  methods  have  been  introduced  to  gain  simple  paths  to 
results,  every  effort  being  made  to  present  a  full  physical  con- 
ception of  phenomena  to  the  reader's  mind.  The  mathematics 
used  are  merely  a  means  to  the  end,  and  are  by  no  means  to  be 
considered  from  any  other  standpoint.  In  this  respect  it  has 
been  sought  to  avoid  either  the  error  of  presenting  unneces- 
sary formulas  or  on  the  other  hand  of  giving  results  without 
reasons,  both  of  which  are  fatal  to  the  reader's  true  progress 
as  they  leave  him  with  no  true  physical  conception  of  the 
phenomena  studied.  Numerous  original  demonstrations  of 
the  standard  formulas,  which  it  is  thought  have  some  merit, 
have  been  introduced,  and  a  few  additions  have  been  made 
to  the  nomenclature.  The  most  important  of  the  latter  is 
the  introduction  of  the  term  active  to  represent  the  com- 
ponent of  pressure  or  electromotive  force  in  phase  with  cur- 
rent, and  to  represent  the  working  component  of  current. 
The  introduction  of  this  term  removes  the  inconvenience,  of 
which  Professor  S.  P.  Thompson  bitterly  complains,  caused  by 
the  use  of  the  term  effective  in  the  formally  adopted  meaning 
of  Vmean2. 

Where  the  volume  is  used  as  a  text-book,  and  time  for 
completing  the  full  course  is  not  available,  the  following 
chapters  may  be  omitted  without  interfering  with  the  conti- 
nuity of  the  subject :  Chapters  IV.,  X.,  XII.,  XIII.,  XIV.,  XV., 


vii 


PREFACE. 


and  the  appendices;  but  an  abbreviated  course  in  this. subject 
is  not  to  be  advised  for  electrical  engineering  students. 

The  foot-notes  given  in  the  text  so  fully  acknowledge  the 
authors'  indebtedness  to  other  writers  that  it  is  perhaps  un- 
necessary to  make  further  acknowledgment  in  this  preface 
beyond  the  statement  that  all  standard  publications  have  been 
drawn  from  as  seemed  desirable,  and  that  the  authors  are 
especially  indebted  to  the  delightfully  lucid  expositions  of 
several  French  writers. 

The  proof  of  the  entire  book  has  received,  to  its  great  advan- 
tage, the  careful  reading  of  Professor  George  D.  Shepardson, 
of  the  University  of  Minnesota,  and  several  chapters  have 
been  read  by  Professor  C.  S.  Slichter,  of  the  University  of 
Wisconsin,  to  both  of  whom  we  are  indebted  for  many  valuable 

suggestions. 

THE   AUTHORS. 
May,  1896. 


TABLE   OF   CONTENTS. 


CHAPTER   I. 

PAGE 

THE  ELECTRIC  PRESSURE  DEVELOPED  BY  ALTERNATORS          .        .        I 

Form  of  pressure  curve;  period  and  frequency;  effect  of 
style  of  winding  on  the  pressure;  relative  dimensions  and  out- 
puts of  continuous-current  and  alternating-current  machines; 
proportion  of  armature  surface  occupied  by  wire. 


CHAPTER   II. 

ARMATURE  WINDINGS  FOR  ALTERNATORS        .      ",~       ...      15 

Classification  of  armatures;  forms  of  windings;  winding  dia- 
grams; drum  armatures;  ring  armatures;  disc  armatures;  pole 
armatures;  collectors;  inductor  alternators;  insulation;  arma- 
ture cores. 


CHAPTER   III. 

SELF-INDUCTION  AND  CAPACITY        .  '     .        .        v     "". '"". """'".      39 

Self-induction  defined;  inductive  pressure;  active  pressure; 
impressed  pressure;  phase;  angle  of  lag;  triangle  of  electric 
pressures;  self-inductance  defined;  the  henry;  examples;  en- 
ergy of  self-induced  magnetic  field;  curves  of  rising  and  falling 
current;  divided  circuits;  shunted  ballistic  galvanometer; 
power  expended  in  inductive  circuit;  time  constant;  examples; 
alternating  current  in  inductive  circuit;  reactance;  impedance; 
inductive  circuits  in  parallel;  resolution  of  irregular  alternat- 
ing-current curves;  effect  of  capacity;  capacity  pressure; 
energy  of  charged  condenser;  curves  of  charge  and  discharge; 
alternating  current  in  circuit  with  capacity;  effect  of  self-induc- 
tance and  capacity  combined;  methods  for  measuring  self-indue- 


CONTENTS. 


tance;  effect  of  varying  permeability;  power  in  alternating-cur- 
rent circuit;  wattmeter;  measuring  angle  of  lag;  power  loops; 
power  factor;  active  current;  wattless  current;  induction  factor; 
methods  for  measuring  power;  inductive  spark;  self-inductance 
of  parallel  wires;  impedance  factor;  distribution  of  current  in 
wires;  skin  effect. 


CHAPTER   IV. 

GRAPHICAL  AND  ANALYTICAL  METHODS  OF  SOLVING  PROBLEMS  IN 

ALTERNATING-CURRENT  CIRCUITS 151 

Graphical  method  explained;  phase  diagram;  vector  dia- 
gram; application  to  series  circuits;  application  to  parallel 
circuits;  application  to  combined  circuits;  analytical  method; 
examples. 

CHAPTER  V. 

THE  MAGNETIC  CIRCUIT  OF  ALTERNATORS 221 

Losses  in  alternators;  armature  ventilation;  radiating  sur- 
face; current  density;  magnetic  leakage;  determination  of 
number  of  armature  conductors;  armature  self-inductance;  cal- 
culation of  field  windings;  armature  reactions;  separate  excita- 
tion; self-excitation;  compound  excitation. 

CHAPTER  VI. 

CHARACTERISTICS,  REGULATION,  ETC 266 

Curve  of  magnetization;  effect  of  armature  reactions  on  ex- 
citing current;  external  characteristic;  loss  line;  measuring 
instruments;  curve  of  magnetic  distribution;  curves  of  press- 
ure and  current;  tracing  curves  of  pressure  and  current;  con- 
tact makers;  areas  of  successive  curves;  effective  values  from 
rectangular  and  polar  curves. 

CHAPTER   VII. 

REGULATION  AND  COMBINED  OUTPUT      .        .        .        ...     313 

Constant-pressure  regulation;  constant-current  regulation; 
synchronism  and  step;  alternators  in  series;  alternators  in  par- 


CONTENTS.  xi 

PAGE 

allel;  synchronizing;  practice  in  parallel  operation;  effect  of 
frequency;  effect  of  armature  inductance;  effect  of  form  of 
pressure  curve;  effect  of  varying  the  excitation;  effect  of  irreg- 
ular angular  velocity. 

CHAPTER  VIII. 

EFFICIENCIES,  ETC.  .        .        .        .  .        .        .        .    370 

Testing  alternators;  shop  tests;  wattmeters  on  high-pressure 
circuits;  output  vs.  efficiency,  weight,  and  cost;  single-phase; 
polyphase;  armature  reactions  of  polyphasers;  connecting 
polyphase  armatures. 

CHAPTER  IX. 

MUTUAL  INDUCTION          ,:  -    ,      •.'...'.        .        .    396 

Transformers;  mutual  induction;  mutual  inductance;  energy 
of  mutual  induction;  transfer  of  electricity  by  mutual  induc- 
tion; primary  and  secondary  coils;  measurement  of  mutual 
inductance;  effect  of  iron  cores;  mutual  induction  of  distribut- 
ing circuits. 

CHAPTER   X. 

OPERATION  OF  IDEAL  TRANSFORMER  AND   EFFECT  OF  IRON  AND 

COPPER  LOSSES          ....        -        •        •        •        •        •        •    426 

Ratio  of  transformation;  magnetic  leakage;  exciting  current; 
core  magnetization;  inherent  regulation  of  ideal  transformer; 
effect  of  losses  on  regulation;  effects  of  self-inductance  and 
capacity;  graphical  method;  transformation  from  constant 
pressure  to  constant  current;  effect  of  hysteresis  and  foucault 
currents  on  exciting  current. 

CHAPTER   XL 

EFFICIENCY  AND  LOSSES  IN  TRANSFORMERS  ,-":.:      ".    r    .    461 

Core  losses;  magnetic  densities;  copper  losses;  radiating 
surface;  current  densities;  testing  transformers;  all-day  effi- 
ciencies; full-load  efficiencies;  weight  efficiencies;  separation 
of  core  losses;  Ford's  tests;  Fleming's  tests. 


xii  CONTENTS. 


CHAPTER   XII. 

PAGE 

DESIGN  OF  TRANSFORMERS       .        .        *       -.        .        .        .        .    513 

Effect  of  frequency;  effect  of  form  of  pressure  curve;  trans- 
former calculations;  example;  joints  in  cores;  ageing  of 
cores;  current  rushes;  impedance  coils;  compensators; 
boosters. 

CHAPTER   XIII. 

POLYPHASE   CONDUCTING    SYSTEMS    AND   THE    MEASUREMENT   OF 

POWER  IN  POLYPHASE  CIRCUITS 546 

Polyphase  conducting  systems;  star  connection;  mesh  con- 
nection; uniform  power  in  polyphase  circuits;  relations  between 
currents  and  pressures;  mutual-  and  self-induction  in  circuits; 
measurement  of  power  in  two-phase  circuits;  measurement  of 
power  in  three-phase  circuits;  method  with  three  wattmeters; 
method  with  two  wattmeters;  method  with  one  wattmeter; 
proof  of  generality  of  method  with  two  wattmeters;  effect  of 
lag  on  wattmeter  readings. 


CHAPTER   XIV. 

ALTERNATING-CURRENT  MOTORS 

Synchronous  motors ;  effect  of  field  strength  on  the  working 
of  synchronous  motors;  graphical  illustrations;  relation  of  arma- 
ture current  to  excitation;  breaking-down  load;  experiments  of 
Bedell  and  Ryan;  induction  motors;  rotating  magnetic  field; 
resultant  magnetizing  power;  action  of  short-circuited  armature; 
uniformity  of  rotating  field;  definitions  of  armature  and  field; 
induction  motors  are  truly  transformers;  magnetizing  current; 
motor  speeds  and  slip;  graphical  illustrations;  torque  of  ideal 
motor;  effect  of  magnetic  leakage;  maximum  torque;  starting 
torque;  maximum  load;  forms  of  armature  windings;  squirrel- 
cage;  independent  short-circuited  coils;  coils  short-circuited  in 
common  ;  field  windings;  relation  of  speed  to  frequency  and 
number  of  poles;  relative  core  losses  in  fields  and  armature; 
starting  and  regulating  devices;  resistances  in  fields;  resistances 
in  armature;  commutated  armature;  effect  of  rotating  field  on 
field  windings;  formulas  derived  from  those  of  the  transformer; 


CONTENTS.  xiii 

PAGE 

exciting  current;  field  ampere-turns;  slip  and  armature  pressure; 
principles  of  design;  constants  for  determining  magnetic  densi- 
ties; formulas  for  field  current;  current  density;  radiating  sur- 
face; resistance  of  armature;  computation  of  losses;  efficiencies; 
power  factor  and  torque;  relation  of  output  to  constructive  details; 
electro-magnetic  repulsion;  single-phase  induction  motors;  reso- 
lution of  alternating  field;  formula  for  single-phase  induction  mo- 
tors; starting  single-phase  motors;  testing  induction  motors; 
direct  measurement;  stray  power  method;  power  factor;  regu- 
lation and  torque;  illustrations;  weight  efficiency;  effect  of 
frequency;  miscellaneous  forms  of  induction  motors;  Stanley 
motor;  condenser  to  supply  wattless  current;  Shallenberger 
meter  ;  Diamond  meter  ;  Ferranti  meter  ;  Thomson  meter  ; 
Monocyclic  system;  effect  of  form  of  pressure  curve;  reversing 
induction  motors. 

CHAPTER   XV. 

POLYPHASE  TRANSFORMERS    .        .        .        .        .        .        .        .     683 

Stationary  transformers  for  polyphase  circuits;  economy  in 
the  use  of  special  polyphase  transformers;  transformation  of 
phases  ;  rotary  transformers  ;  connecting  the  armature  wind- 
ings; ratio  erf  transformation;  capacity. 


APPENDICES. 

A.  THE    APPLICATION    OF    FOURIER'S  SERIES    TO    ALTERNATING- 

CURRENT  CURVES 695 

B.  THE    CHARACTERISTIC    FEATURES    OF    ALTERNATING-CURRENT 

CURVES .        .        ..'."'.     703 

C.  OSCILLATORY  DISCHARGES  .        .     : •     .     703 

D.  ELECTRICAL  RESONANCE     .        .        .        .        .        .        .        .     709 


INDEX 


719 


LIST    OF    IMPORTANT    SYMBOLS. 


A,  area,  constants. 

fi,  magnetic  induction  per  sq.  cm.,  or  magnetic  density. 

J3m,  maximum  magnetic  density. 

Ba,  magnetic  density  in  armature  core. 

Bf,  magnetic  density  in  field  core. 

C,  electric  current,  effective  value  of  alternating  current. 
c,  instantaneous  current. 

cm,  maximum  value  of  alternating  current. 

C1,  primary  current. 

C",  secondary  current.      |  In   chapters   dealing   with    trans- 

Ci'j  exciting  current.  formers  and  induction  motors. 

CM,  magnetizing  current,  j 

D,  d,  readings  of  instruments. 

E,  electric  pressure  or  electromotive  force,  effective  value  of 

alternating  pressure. 

<?,  instantaneous  electric  pressure. 

em,  maximum  value  of  alternating  pressure. 

EM  active  pressure  (effective  value) . 

Eit  impressed  pressure  (effective  value). 

Est  reactive  pressure  (effective  value). 

F,  friction  loss  in  watts. 
/,  frequency. 

G,  galvanometer  resistance. 

Ht  magnetic  force,  hysteresis  loss  in  watts. 

7,  impedance. 

XV 


xvi  LIST   OF   IMPORTANT   SYMBOLS. 

X,  k,  constants. 

k,        ratio  of  transformation   in    transformers   and   induction 

motors. 
Z,       self-inductance  (in  a  few  efficiency  formulas  L  is  taken 

to  mean  losses). 

Z',      primary  self-inductance.      1  In  transformers  and  induc- 
Z".     secondary  self-inductance.  J       tion  motors. 
/,         length. 

M9      mutual  inductance,  magneto -motive  force. 
m,       number  of  phases  in  polyphase  circuits. 
N,      number   of  lines   of  force, 


or  total  magnetism. 
Na,     armature  magnetism. 


f 


Maximum    magnetism   in 
transformers  and  induc- 


Nf,     field  magnetism.  tion  motors. 

Nlt     leakage  magnetism. 

n,        number  of  turns  in  a  coil. 


,  Transformers  and   induc- 
n,       number  of  primary  turns. 

tion  motors. 
n",      number  of  secondary  turns.  J 

nc,  ampere-turns. 

P,  magnetic  reluctance. 

p,  number  of  pairs  of  poles. 

Q,  q,  quantity  of  electricity. 

qt  instantaneous  quantity  of  electricity. 

R,  r,  resistance. 

S,  number  of  armature  conductors. 

S',       number  of  primary  conductors. 

"  induction  motors. 


, } In " 


S",  number  of  secondary  conductors. 

s,  capacity  of  condenser. 

T,  t,  time. 

/,  instant  of  time. 

U,  periphery  velocity. 

V,  revolutions  per  minute. 

V9  instantaneous  velocity,  slip  in  induction  motors. 


LIST   OF   IMPORTANT   SYMBOLS.  xvii 

Vi,  frequency  of  current  in  armature  of  induction  motor. 

Wt  power. 

w,  instantaneous  power. 

Z,  foucault  current  loss  in  watts. 

z,  leakage  coefficient. 

a,  angles,  phase  of  alternating  current. 

ft,  angles  or  deflections. 

8,  deflections. 

e,  base  of  Napierian  logarithms. 

r),  commercial  efficiency. 

0,  angles  or  deflections. 

p,  magnetic  permeability. 

TT,  ratio  of  circumference  to  radius. 

T,  time  constant. 

<£,  angle  of  lag. 

\f/,  angle  between  currents  C'  and  Ci  in  transformers. 

w,  angular  velocity. 


NOTE.  —  Students  who  are  acquainted  with*  the  methods  of  solving 
differential  equations  may  pass  over  the  solution  of  the  equation 


given  in  Section  24.     This  is  a  linear  equation  of  the  first  order,  and  its 
solution  may  be  written  down  at  once. 

Students  who  are  not  acquainted  with  the  elementary  principles  of 
complex  quantities  may  advantageously  read  Chapter  31  of  Van  Velzer 
and  Slichter's  University  Algebra  before  entering  upon  Section  56. 


ALTERNATING  CURRENTS. 

CHAPTER   I. 

THE  ELECTRIC  PRESSURE  DEVELOPED  BY  ALTERNATORS. 

IN  dealing  with  alternating  currents,  several  variables 
enter  into  the  problem  which  make  it  impossible  to  use, 
without  modification,  the  results  gained  in  the  study  of 
continuous  currents.  But  the  same  fundamental  laws 
control  the  phenomena  of  both,  and  if  care  is  taken 
to  apply  these  with  due  regard  to  the  limiting  condi- 
tions, it  will  be  found  that  the  subject  may  be  clearly 
grasped  and  be  reduced  almost  to  the  simplicity  of 
continuous-current  work.  In  many  particulars,  the 
design  of  alternating-current  machinery  varies  widely 
from  that  of  continuous-current  apparatus,  as  the  re- 
quirements are  materially  different,  though  in  the  mat- 
ter of  good  workmanship  and  substantial  construction 
there  is  no  difference  in  the  two  classes  of  machines. 

1.  Form  of  the  Pressure  Curve  of  an  Alternator.  — 
In  a  multiple-coil  continuous-current  armature,  if  5 
be  the  number  of  conductors,  Na  the  useful  magnetic 
lines  of  force  passing  into  the  armature,  and  V  the 


2  ALTERNATING   CURRENTS. 

speed  in  revolutions  per  minute,  the  pressure  developed 

willbe  £=  SN.V  * 

io8x6o 

In  the  demonstration  in  Vol.  I.  it  has  been  assumed 
that  the  magnetic  field  is  uniform,  which  is  by  no  means 
the  case  in  commercial  dynamos.  The  assumption,  how- 
ever, simplifies  the  demonstration  without  introducing 
error  into  the  result,  as  will  now  be  shown. 

Suppose  the  field  be  perfectly  uniform,  then  each 
armature  conductor  will  cut  lines  of  force  at  the  uni- 
form rate  of  —  = —  lines  per  second.  If  the  field 

dt  60 

be  not  uniform,  each  conductor  will,  at  any  instant,  cut 
lines  of  force  at  the  rate  — ,  which  will  be  variable, 

but   the   average   value    of   which   is   evidently — 

60 

Since  the  conductors  follow  each  other  in  consecutive 
order,  and  the  sum  of  the  lines  cut  by  all  the  con- 
ductors at  each  instant  is  practically  uniform  through- 
out the  revolution,  the  total  electric  pressure  developed 
must  be  proportional  to  the  average  rate  of  cutting  by 
each  conductor  multiplied  by  the  number  of  conductors 
in  series.  Or,  as  above, 


io8x6o 

In  the  case  of  alternating-current  dynamos  this  aver- 
aging does  not  occur,  because  the  number  of  lines ; 
which  are  cut  at  each  instant  by  all  the  conductors  in 
series  is  variable.  Thus,  assume  that  in  Figs,  i  and  2! 
(reproduced  from  Fig.  36,  Vol.  I.),  the  field  is  altered  so 

*  Textbook,  Vol.  I.  Chap.  4. 


ELECTRIC   PRESSURE   DEVELOPED.  3 

that  it  is  weakest  near  the  centre  ;  then  the  rate  of  cut- 
ting lines  no  longer  varies  as  a  sine  curve,  but  the  curve 


Fig.  1 


Fig.  2 


rises  more  rapidly  in  its  lower  portions  and  does  not 
reach  as  great  a  maximum.    If  the  field  be  concentrated 


4  ALTERNATING   CURRENTS. 

towards  the  centre,  the  curve  becomes  higher  at  the 
centre,  but  does  not  rise  as  rapidly  in  its  lower  portions. 
Figure  3  shows  the  curve  of  electric  pressure  developed 
in  a  coil  which  revolves  in  a  field  of  the  same  total 
number  of  lines  of  force  as  that  of  Figs,  i  and  2,  but  in 
which  the  induction  at  the  centre  is  50  per  cent  less 
than  the  average.  Figure  4  shows  a  similar  curve 
when  the  induction  at  the  centre  is  50  per  cent  greater 
than  the  average.  In  each  case,  the  induction  is  as- 


Fig.  3 


sumed  to  change  gradually  and  uniformly  from  the 
centre  to  the  edges.  In  all  cases  where  a  certain  total 
number  of  lines  is  cut  per  revolution  by  a  coil  revolving 
at  constant  speed,  the  average  electric  pressure  remains 
constant  regardless  of  the  magnetic  distribution,  but  the 
effective  pressure  (VavT?)  is  by  no  means  independent 
of  the  distribution.  Taking  the  maximum  pressure  of 
the  sine  curve  shown  in  Fig.  2  as  the  unit,  the  average 

2 

pressure  in  each  figure  is  —  =  .637  (see  p.  82,  Vol.  I.). 

7T 

The  effective  pressure  of  the  sine  curve  shown  in  Fig.  2 


ELECTRIC   PRESSURE   DEVELOPED. 


is  -  —  =  .707.     The  maximum  pressures  shown  in  the 

V2 

curves  of  Figs.  3  and  4  are  .81  and  1.5,  and  the  effective 
pressures  are  .58  and  .85.  If  the  field  were  distributed 
as  in  Fig.  5  (the  total  magnetization  remaining  constant), 


Fig.  4 


the  maximum,  average,  and  effective  pressures  would  be 
equal  to  each  other  (Fig.  6),  and  of  a  numerical  value 
of  .637.  On  the  other  hand,  if  the  field  be  greatly  con- 
centrated towards  the  centre,  as  in  Fig.  7,  the  maximum 
pressure  is  very  great,  and  the  effective  pressure  is  con- 


ALTERNATING   CURRENTS. 


siderably  greater  than  the   average,   though   it  by  no 
means  approaches  in  value  the  maximum  pressure. 


Fig.  5 

2.    Period  and  Frequency  of  an  Alternating  Current. — 

The  time   in   seconds,    T,   required  to  pass  through  a 
complete  cycle  or  curve  is  called  the  Period  of  an  alter- 


180' 


360° 


Pig.  6 

nating  electric  current  or  pressure.  The  number  of 
periods  in  a  second  is  called  the  Frequency  of  the  cur- 
rent or  pressure  (this  term  was  adopted  by  the  Paris 


ELECTRIC   PRESSURE   DEVELOPED.  / 

Electrical  Congress).  Usually  an  alternating  current 
or  pressure  is  designated  by  its  effective  value  and  fre- 
quency. It  is  quite  common,  however,  to  use  the  num- 
ber of  half-periods,  or  the  number  of  Alternations  in  a 
minute,  instead  of  the  frequency.  In  this  case,  the 
number  of  alternations  is  equal  to  2  x  60  x  the  fre- 
quency. Example:  a  current  with  a  frequency  of  100 
makes  12,000  alternations  per  minute.  The  term  Peri- 
odicity is  sometimes  used  for  frequency. 


Fig-.  7 

For  the  general  commercial  purposes  of  the  present 
day  the  frequency  of  alternating  currents  varies  widely, 
but  nearly  all  cases  fall  within  the  limits  of  40  and 
135  (4800  and  16,200  alternations  per  minute).  The 
majority  of  American  alternating-current  dynamos,  or 
Alternators,  give  a  frequency  of  between  120  and  135, 
but  in  Europe  a  somewhat  lower  frequency  is  more 
common,  and  a  frequency  of  60  is  coming  into  quite 
general  use  in  this  country.  The  rotation  of  a  coil  in  a 


8  ALTERNATING   CURRENTS. 

two-pole  field  gives  one  complete  period,  or  two  alterna- 
tions, for  each  revolution,  and  the  frequency  is  therefore 
equal  to  the  number  of  revolutions  per  second.  Since 
the  armatures  of  two-pole  machines  would  be  required 
to  run  at  an  impracticable  speed  in  order  to  give  the 
ordinary  commercial  frequencies,  alternators  are  nearly 
always  made  with  a  considerable  number  of  poles.  The 
number  of  poles  depends  upon  the  size  of  the  alternator 
and  other  conditions  which  may  control  the  speed  of  the 
armature,  but  in  general,  it  may  be  said  to  vary  from 
eight  upwards.  A  great  many  of  the  alternators  built 
in  the  United  States  have  been  designed  to  give  16,000 
alternations  per  minute  at  a  speed  of  1600  revolutions 
per  minute,  and  hence  have  been  built  with  ten  poles. 
The  frequency  which  is  produced  by  an  alternator  is 
either  equal  to  one-sixtieth  of  the  product  of  the  num- 
ber of  its  pairs  of  poles  and  the  speed  of  its  armature 
in  revolutions  per  minute,  or  is  twice  as  great.  The 
number  of  alternations  per  minute  is  equal  to  the  fre- 
quency multiplied  by  120,  as  shown  above. 

3.  Field  Excitation  of  Alternators.  —  As  an  alternat- 
ing current  will  not  serve  to  magnetize  the  fields  of 
dynamos,  some  arrangement  for  obtaining  a  direct  cur- 
rent  must   be   made  for  the  excitation   of  alternators. 
This  may  be  done  either  by  commutating  or  rectifying 
all  or  a  part  of  the  alternating  current  produced  by 
the  machine,   or   a   small  auxiliary  continuous-current 
dynamo  called  an  Exciter  may  be  supplied  for  the  pur- 
pose.     Sometimes  the  exciter  is  mounted  on  the  bed 
plate  of  the  alternator  (compare  Sect.  71). 

4.  Effective  Pressure.  —  Assuming  that  the  armature 


ELECTRIC   PRESSURE   DEVELOPED.  9 

conductors  are  so  arranged  that  all  may  be  effective,  it 
is  seen  from  what  has  preceded  that  the  average  press- 

ure  developed    by   an   alternator   is  -  —  5  —  ^—  ^,    where 

io8  x  60 

Sr  is  the  number  of  conductors  in  series,  Na  is  the 
number  of  lines  of  force  passing  through  the  arma- 
ture in  each  magnetic  circuit  (emanating  from  each 
pole),  and  p  is  the  number  of  pairs  of  poles.  If  the 
windings  of  the  armature  are  all  connected  up  in  series, 

as  is  common,  this  becomes  —  5  —  l~J£.  where  S  is  the 

io8  x  60 

number  of  conductors  on  the  armature.  From  p.  278, 
Vol.  I.,  it  is  seen  that  in  multipolar  continuous-current 
dynamos  with  series-path  armatures, 


= 

io8  x  60' 

Now,  assuming  two  machines,  one  producing  continuous 
currents  and  the  other  alternating  currents,  in  which 
S,  Na,  V,p,  are  equal,  the  formulas  show  that  the  average 
electric  pressure  developed  in  the  alternator  armature 
with  its  conductors  all  in  series  is  twice  that  developed 
in  the  continuous-current  armature  with  halves  in  par- 
allel. The  effective  alternating  pressure  of  commer- 
cial machines  has  been  shown  to  be  usually  greater  than 
the  average  pressure.  Calling  the  ratio  of  effective  to 
average  pressure  k,  it  is  seen  that  the  effective  press- 
ure of  the  alternator  with  armature  conductors  all  in 
series  is  2k  times  that  of  the  continuous-current  ma- 
chine. On  the  other  hand,  in  the  continuous-current 
armature  two  paths  exist  for  the  current  against  one 
in  the  alternator  armature.  Consequently,  the  out- 


10  ALTERNATING   CURRENTS. 

put  of  the  alternator  armature  is  only  k  times  greater 
than  that  of  the  continuous-current  armature.  The 
value  of  k  when  the  current  curve  is  sinusoidal  is 

I  2 

— = -5-— =1.11.      Hence  the  electric  pressures  are  to 

V2        T 

each  other  as  I  :  2.22,  and  the  outputs  are  in  the  ratio 
of  1:1.11.  The  windings  on  alternator  armatures  are 
frequently  connected  so  that  the  two  halves  are  in 
parallel,  instead  of  in  series,  in  which  case  the  pressure 
developed  with  a  given  number  of  conductors  is  halved, 
but  the  current  capacity  is  at  the  same  time  doubled. 
In  this  case  both  the  pressures  and  outputs  of  the  con- 
tinuous- and  alternating-current  machines  have  the  ratio 
of  i  :  1. 1 1.  The  value  of  k  in  commercial  alternators 
depends  upon  the  ratio  which  the  width  of  the  poles 
bears  to  the  pitch  (distance  between  poles,  centre  to 
centre)  and  upon  the  relative  arrangement  of  the  wind- 
ings. It  has  been  shown  (Sect,  i)  to  have  a  minimum 
limit  of  unity  and  a  maximum  limit  which  may  be  very 
great.  In  commercial  machines  it  may  be  said  to  gen- 
erally lie  between  about  i  and  1.25. 

5.  The  Effect  of  Differential  Action  by  the  Armature 
Conductors.  —  It  is  now  well  to  examine  the  relative 
sizes  of  the  two  armatures  compared  above,  and  the 
effect  of  the  arrangement  of  the  windings  upon  the 
pressure  developed.  Thus  far  the  alternator  windings 
have  been  assumed  to  be  in  narrow  coils,  or  arranged 
so  that  the  conductors  are  equally  effective  at  every 
instant.  This  cannot  be  the  case  in  actual  machines, 
as  the  coils  must  have  appreciable  width.  If  two  col- 
lecting rings  are  placed  upon  the  shaft  of  a  contin- 


ELECTRIC  PRESSURE  DEVELOPED.       II 

uous-current  armature,  such  as  a  Gramme  ring,  and  con- 
nected to  armature  conductors  which  are  in  opposite 
coils,  an  alternating  current  may  be  taken  from  the 
rings.  The  electric  pressure  between  the  rings  has  its 
maximum  when  the  conductors  connected  to  the  rings 
are  under  the  points  of  commutation,  and  the  pressure 
is  zero  when  the  armature  has  revolved  90°  (compare 
Vol.  I.,  p.  90).  (In  the  case  of  a  narrow  coil,  the  latter 
point  is  the  point  of  maximum  pressure?)  The  maximum 
of  the  alternating  pressure  must  be  equal  to  the  con- 
tinuous pressure  for  which  the  armature  was  designed, 
and  the  effective  alternating  pressure  is  .707  times  the 
continuous  pressure  if  the  field  is  uniform.  In  com- 
mercial machines  the  field  is  not  uniform,  but  it  is  not 
likely  to  be  sufficiently  irregular  to  materially  disturb 
the  ratio  of  the  continuous  and  effective  alternating 
pressures  when  the  armature  is  carrying  little  current. 
When  the  armature  carries  considerable  current,  arma- 
ture reactions  may  disturb  the  relations  to  a  greater 
or  less  degree.  An  armature  thus  arranged,  with  the 
conductors  uniformly  distributed  over  its  surface,  gives 
a  sine  alternating  current.  The  value  of  k  is  therefore 
1. 1 1  ;  but  the  effective  alternating  pressure  is  only  .707 
times  that  of  the  continuous  pressure  developed  by  the 
same  conductors.  The  question  at  once  arises  as  to 
the  cause  of  this  loss  of  40  per  cent  or  more.  A  little 
consideration  shows  that  the  Gramme  ring  acts  like  two 
broad  coils  in  parallel,  which  cover  the  whole  armature 
and  unite  at  the  points  where  the  collecting  rings  are 
connected.  When  in  the  position  of  zero  pressure 
the  two  halves  of  each  coil  are  so  located  in  the  field 


12  ALTERNATING   CURRENTS. 

as  to  cut  lines  in  opposite  directions,  and  hence  the 
pressure  is  zero.  Similar  differential  action  is  found 
when  the  coils  cover  only  a  portion  of  the  armature,  the 
extent  of  the  effect  depending  upon  the  ratio  of  the 
width  and  pitch  of  the  poles  to  the  width  of  the  coils. 
It  is  almost  entirely  avoided  if  the  coils  are  never  wider 
than  the  distance  between  the  pole  tips.  On  account 
of  this  differential  action,  it  is  necessary  to  include 
another  constant  in  the  formula  for  the  electric  press- 
ure developed  by  alternating-current  armatures.  Call- 
ing this  constant  k' ,  and  replacing  the  product  k'k  by 

JS      4-U        t  1        U  77         2KSNaVp 

A,  the  formula  becomes  E  = 5 — •—*-.    Kapp  states 

io8  x  60 

that  K  varies  from  .29  to  1.15,*  but  in  the  greater 
number  of  commercial  cases  it  is  between  i.oo  and 
i. ii. 

The  output  of  an  alternator  is  proportional  to  the 
product  SNa  —  swM,  where  s  is  the  number  of  conduct- 
ors per  unit  width  of  coil,  b  the  number  of  lines  of 
force  per  unit  width  of  pole  face,  and  w  and  w1  are 
respectively  the  widths  of  coil  and  pole  face.  In  order 
to  economize  material,  the  distance  between  pole  tips 
may  be  taken  as  equal  to  the  width  of  coil,  and  this 
makes  w  +  w'  equal  to  the  pitch  of  the  poles,  which  is 
constant ;  this  makes  the  product  wwf,  and  hence  the 
output  of  the  machine,  a  maximum  when  w  is  equal  to 
w' ,  or  when  the  width  of  coil  and  pole  face  are  each 
equal  to  half  the  pitch.  The  result  thus  derived  must 
be  modified  to  suit  practical  conditions,  since  fringing 
tends  to  increase  the  width  of  field,  and  armature  reac- 

*  Kapp's  Dynamos,  Alternators,  and  Transformers,  p.  374- 


ELECTRIC  PRESSURE  DEVELOPED.       13 

tions  tend  to  crowd  the  field  towards  the  trailing  pole 
tip,  thus  narrowing  the  field.  Experiment  has  shown 
that  it  is  best  to  have  the  coils  somewhat  wider  than 
half  the  pitch,  and  it  is  often  advantageous  to  have  the 
poles  of  slightly  less  width.* 

6.  Relative  Dimensions  of  Continuous-  and  Alternating- 
Current  Dynamos.  —  From  this  discussion  it  is  seen  that 
only  about  one-half  the  surface  of  alternator  armatures 
should  be  covered  with  wire,  so  that  the  construction 
is  quite  different  from  that  of  continuous-current  arma- 
tures. With  a  fixed  size  of  armature  and  depth  of 
winding  it  is  apparent  that  an  alternating-current  arma- 
ture will  carry  only  about  half  as  much  wire  as  is  carried 
by  a  similar  continuous-current  armature.  In  order  to 
gain  an  equal  output  from  the  machines,  it  is  therefore 
necessary  to  increase  either  the  size  of  the  alternator 
armature  or  its  speed.  Both  of  these  devices  are  used 
in  common  practice,  and  it  is  not  unusual  for  an  alter- 
nator armature  to  be  run  at  a  surface  speed  upwards  of 
7000  feet  per  minute.  The  attached  table  shows,  to 
some  degree,  how  far  the  effect  of  the  difference  in  the 
lengths  of  the  wire  on  the  armatures  is  overcome  in 
practice.  In  the  comparison,  it  must  be  remembered 
that  the  induction  in  the  alternator  is  usually  considera- 
bly less  than  that  in  continuous-current  machines  (com- 
pare Sect.  61). 

*  Compare  Elihu  Thomson  in  Proc,  Institution  of  Civil  Engineers, 
Vol.  97,  p.  loi. 


ALTERNATING   CURRENTS. 


ALTERNATORS. 


Designa- 
tion. 

Revolutions 
per  Minute. 

Diameter 
rev.  part. 

Periphery 
Speed. 

Frequency. 

Conductor. 
Inches 
per  Volt. 

Capacity 
in  K.W. 

I 

1200 

18" 

5700 

IOO 

8 

15 

2 

600 

60 

14-5 

30 

3 

I5OO 

18" 

7100 

»5 

14 

35 

4 

1500 

18" 

7IOO 

I25 

9 

35 

5 

22 

40 

6 

400 

52" 

5400 

86 

40 

7 

500 

83 

II.  2 

5° 

8 

1650 

18" 

7800 

138 

5° 

9 

600 

48" 

7500 

70 

15 

60 

10 

600 

42" 

6600 

IOO 

9.6 

1  20 

ii 

I080 

24" 

6500 

144 

15 

140 

12 

335 

66" 

5800 

67 

15.2 

225 

CONTINUOUS-CURRENT  DYNAMOS. 


Designa- 
tion. 

Revolutions 
per  Minute. 

Diameter  of 
Armature. 

Periphery 
Speed. 

Conductor. 
Inches 
per  Volt. 

Capacity  in 

K.W: 

j 

1750 

6.25 

2850 

IO 

6 

2 

2750 

4-5 

3200 

n-5 

6 

3 

800 

10 

2200 

24 

7-5 

4 

460 

'7-5 

2250 

26 

20 

5 

900 

12.5 

3OOO 

18 

35 

6 

1650 

12 

3300 

15 

35 

7 

800 

10 

2IOO 

!3 

35 

8 

530 

H 

195° 

18 

5° 

9 

700 

15-5 

2850 

18 

55 

10 

900 

16 

3800 

18 

60 

ii 

4OO 

15.5 

I6OO 

26 

70 

12 

45° 

*7-5 

2050 

20 

80 

13 

400 

25 

26OO 

19 

150 

ARMATURE  WINDINGS  FOR  ALTERNATORS.       15 


CHAPTER    II. 

ARMATURE   WINDINGS    FOR    ALTERNATORS.        .. 

7.  Classification  of  Armatures.  —  Referring  to  page 
93  of  Volume  I.  it  will  be  seen  that  dynamo  armatures 
are  classified  under  three  divisions  : 

1.  Where  a  wire  cuts  lines  of  force  by  moving  across 
them,  as  in  the  case  of  a  slider  or  of  a  wire  moving 
around  a  magnet  pole. 

2.  Where  a  coil,  or  set  of  coils,  is  moved  parallel  to 
itself,   or  nearly  so,   between  points  of  different   field 
strength. 

3.  Where  a  coil,  or  set  of  coils,  is  wound  on  a  ring 
or  drum  and  given  a  rotary  motion  in  a  fixed  magnetic 
field. 

The  first  division  includes  only  the  so-called  unipolar 
armatures,  and  requires  no  treatment  here.  Alternator 
armatures  are  in  general  classed  in  the  second  division. 

It  is  not  uncommon  for  the  field  to  move  instead  of 
the  armature,  and  in  some  cases  neither  the  armature 
nor  field  coils  move,  but  the  induction  through  the 
former  is  varied  by  the  revolution  of  iron  Inductors. 
Where  the  field  rotates,  or  the  variation  of  the  induc- 
tion is  effected  by  moving  inductors,  the  electric  press- 
ure generated  in  the  armature  windings  is  produced 
in  exactly  the  same  manner  as  if  they  moved,  and 
such  armatures  therefore  belong  to  the  second  division. 


16  ALTERNATING   CURRENTS. 

The  construction  of  the  machines  thus  enumerated 
requires  an  additional  classification  into : 

1.  Alternators  with  moving  armatures. 

2.  Alternators  with  moving  fields. 

3.  Inductor  alternators. 

The  armatures  of  alternators  belonging  to  the  first 
and  second  of  these  classes  may,  for  convenience,  be 
divided  into  four  classes  : 

1.  Drum  armatures.  3.    Disc  armatures. 

2.  Ring  armatures.  4.    Pole  armatures. 

8.  Drum  Armatures.  —  Drum-wound  alternator  arma- 
tures are  always  chord-wound  (see  p.  273,  Vol.  I.).  In 
two-path  chord-wound  continuous-current  drum  arma- 
tures, the  individual  turns  of  each  armature  section 
cross  each  other  on  the  heads  in  very  much  the  same 
way  as  in  armatures  for  two-pole  machines  (Fig.  8). 
An  exactly  similar  arrangement  may  be  used  in  wind- 
ing alternating-current  armatures,  but  the  armature 
surface  is  not  entirely  covered  with  wire  (Fig.  9). 
Since  either  the  whole  or  one-half  of  the  armature  wire 
is  connected  permanently  in  series  and  only  the  ends  of 
the  windings  are  carried  to  collecting  rings,  there  is  no 
advantage  to  be  gained  by  permitting  the  wires  to 
cross  each  other  in  alternator  armatures.  The  crossing 
of  the  wires,  on  the  other  hand,  presents  a  positive 
disadvantage  by  increasing  the  difficulty  of  satisfac- 
torily insulating  them.  This  is  particularly  marked 
on  account  of  the  high  pressures  for  which  alternat- 
prs  are  commonly  built.  The  winding  of  alternator 


ARMATURE  WINDINGS  FOR  ALTERNATORS.       I/ 


i8 


ALTERNATING   CURRENTS. 


armatures   is    therefore   advantageously   made    without 
the  wires    being  crossed  (Fig.   10).     This  is  called  by 


\ 


\ 


S.   P.  Thompson  *  wave  winding,  but   may  be   prefer- 


*  Thompson's  Dynamo- Electric  Machinery,  4th  ed.,  p.  314. 


ARMATURE  WINDINGS  FOR  ALTERNATORS.       19 

ably  called  undulatory  winding  after  Fritsche,*  or 
better  still,  Continuous  Winding.  The  same  result  may 
evidently  be  obtained  by  turning  the  wires  back  at  the 


ends  of  the  armature  core  so  that  they  form  isolated 
coils  (Fig.  n),  and  then  properly  connecting  the  coils. 
When  there  are  as  many  coils  as  poles,  alternate  coils 


Fritsche's  Die  Gleichstrom  Dynamomachine,  p.  43. 


2O  ALTERNATING   CURRENTS. 

are  moving  at  any  instant  under  poles  of  opposite  signs, 
and  the  electric  pressures  developed  in  them  are  in 
opposite  directions.  They  must  therefore  be  so  con- 
nected that  they  are  alternately  right-  and  left-handed 
(Figs.  1 1  and  1 2) ;  and  likewise  when  there  are  half  as 
many  coils  as  poles,  the  coils  must  all  be  connected  in 
the  same  direction  (Figs,  na  and  12  a).  This  arrange- 


Fig.  12 

ment  may  be  called  Coil  Winding,  or,  after  E.  Arnold  * 

and  S.  P.  Thompson,!  Loop  Winding  or  Lap  Winding.\ 

As  already  stated  (Sect.  4),  alternator  armatures  are 

frequently  connected  with  the  two  halves  of  the  wind- 

*  Die  Ankerwickelungen  der  Gleichstrom-Dynamomachinen,  p.  13. 
f  Dynamo- Electric  Machinery,  4th  ed.,  p.  314. 

J  For   numerous   diagrams   of  alternator   windings   see   Parshall   and 
Hobart's  Armature  Windings,  Chap.  12. 


ARMATURE  WINDINGS  FOR  ALTERNATORS.       21 

ings  in  parallel.  In  this  case,  the  coils  must  be  con- 
nected in  each  half  so  that  they  are  alternately  right- 
and  left-handed,  but  where  the  halves  join,  both  the 
coils  must  be  either  right-  or  left-handed  (Fig.  13). 
When  the  coils  are  all  connected  in  series,  the  first 
and  last  coils  lie  side  by  side ;  consequently  in  arma- 
tures built  for  high  pressures  a  severe  strain  is  thrown 


Fig.  12  a 

upon  the  insulation  separating  them.  When  the  halves 
of  the  armature  are  connected  in  parallel,  the  first  and 
last  coils  lie  on  opposite  sides  of  the  armature,  and 
effective  insulation  is  therefore  less  difficult  of  attain- 
ment. In  this  case,  the  pressure  between  adjacent  coils 
cannot  be  greater  than  the  total  pressure  divided  by 
half  the  number  of  coils.  When  the  coils  are  all  in 


22  ALTERNATING   CURRENTS. 

series  the  pressure  between  any  adjacent  coils,  except- 
ing between  the  two  end  coils,  is  equal  to  the  total 
pressure  divided  by  the  total  number  of  coils.  When 
an  alternator  armature  is  connected  with  its  halves  in 
parallel,  every  precaution  must  be  taken  to  make  the 
pressures  developed  in  the  two  halves  equal  at  every 
instant ;  otherwise  there  is  danger  of  one  portion  of  the 
armature  overpowering  the  other,  exactly  as  is  some- 


Fig-.  13 

times  the  case  in  multipolar  continuous-current   arma- 
tures (Vol.  L,  p.  275). 

The  armature  conductors  may  lie  upon  the  surface  of 
the  core  or  be  imbedded  below  it.  Where  toothed  cores 
are  used,  the  teeth  are  equal  in  number  to  the  number 
of  coils,  and  the  armature  approaches  the  fourth  class 
or  pole  type.  Toothing  increases  armature  reactions 
and  self-inductance,  thus  interfering  with  regulation, 
but  it  is  in  general  quite  advantageous  and  is  coming 


ARMATURE  WINDINGS  FOR  ALTERNATORS.       23 

into  general  use  by  the  better  class  of  manufacturers 
(see  Sect.  n). 

The  coils  for  alternator  drum  armatures  are  ordinarily 
wound  on  formers  and  then  fastened  upon  the  arma- 
tures after  having  been  well  insulated.  Where  surface 
windings  are  used,  the  coils  are  frequently  arranged  to 
bend  down  over  the  ends  of  the  core,  as  in  Fig.  14, 
where  they  are  securely  fastened  by  end  plates,  or  blocks 
of  wood  or  fibre.  It  is  usual  to  fill  the  spaces  in  the 


Pig.  14 

centres  of  the  coils  with  wood  blocks,  which  are  screwed 
to  the  cores  or  are  held  by  the  binding  wires.  (Exam- 
ples :  Westinghouse  and  National  alternators.)  In  some 
machines  the  coils  are  flat,  or  pancake-like,  and  of  the 
same  length  as  the  armature  core.  In  this  case  they 
are  laid  upon  the  cylindrical  surface  of  the  armature 
core  and  securely  bound  with  wire  bands.  (Examples : 
Thomson-Houston  and  Elwell-Parker  alternators.)  The 
high  periphery  velocities  of  alternator  armatures  make 
heavy  bands  essential  to  the  preservation  of  surface 


24  ALTERNATING   CURRENTS. 

windings,  and  all  blocking  must  be  securely  fastened. 
The  wood  blocks  which  fill  the  centre  of  the  coils  make 
excellent  driving  teeth,  and  therefore  serve  a  good  pur- 
pose. When  imbedded  coils  are  used,  they  may  be 
made  upon  formers  (lathe-wound),  and  then  applied  to 
the  core,  or  the  conductors  may  be  threaded  through  the 
grooves,  which  are  well  insulated.  When  the  core  teeth 
are  T-shaped,  the  coils  may  be  wound  of  sufficient 
width  to  slip  over  the  head,  and  when  in  place  they 
may  be  narrowed  by  squeezing.  The  coils  for  this  pur- 
pose must  be  wound  with  ends  of  such  shape  that  they 


Figf.  15 

will  bend  without  injury,  as  is  shown,  for  example,  in 
Fig.  15.  Lathe-wound  coils  are  of  decided  advantage 
for  armatures  designed  for  high  pressures,  as  their  insu- 
lation may  be  made  particularly  safe.  Such  coils  are 
usually  served  with  layers  of  japanned  canvas,  fuller  or 
press  board,  vulcanized  fibre,  and  mica.  The  slots 
between  the  core  teeth  may  be  made  of  sufficient  area 
to  permit  the  use  of  any  desirable  thickness  of  insula- 
tion, and  the  teeth  make  a  very  complete  mechanical 
protection  for  the  coils.  Toothed  armatures  with  lathe- 


ARMATURE  WINDINGS  FOR  ALTERNATORS.       25 

wound  coils  should  therefore  be  thoroughly  reliable.  If 
the  magnetic  surface  of  the  armature  is  fairly  complete, 
the  wires  are  protected  from  magnetic  drag  (p.  153, 
Vol.  I.),  which  decreases  the  chances  of  the  conductors 
chafing  and  therefore  injuring  the  insulation. 

9.  Ring  Armatures.  —  Ring-wound  alternator  arma- 
tures were  early  used  with  commercial  success,  and 
some  of  the  old  Magneto  Machines  of  the  De  Meritens 
type  with  permanent  field  magnets  and  ring  armatures 
are  still  in  operation.  The  invention  of  the  ring  arma- 
ture for  alternating-current  machines  is  usually  ascribed 


Fig.  16 

to  either  Gramme  or  Wilde,  who  independently  patented 
the  form  in  France  and  England  in  1878.  In  Amer- 
ica, ring  armatures  have  not  been  viewed  with  as  great 
favor  as  have  drum  armatures,  probably  on  account  of 
the  smaller  mechanical  stability  in  the  ring  and  the 
greater  self-induction  in  its  windings.  As  in  drum 
armatures,  the  coils  of  ring-alternator  armatures  must 
be  connected  alternately  right-handed  and  left-handed 
(Fig.  1 6).  The  ring  may  be  arranged  so  that  the  field- 
magnets  surround  it,  as  in  Fig.  16;  so  that  it  surrounds 
the  fields,  as  in  Fig.  17,  in  which  case  the  latter  usually 
revolve  (examples  :  Gramme  and  Siemens  &  Halske  al- 
ternators); or  with  the  field  poles  as  a  crown  upon  both 


26 


ALTERNATING   CURRENTS. 


sides  of  it  (example :  Kapp  alternator).  In  the  latter 
case,  the  ring  may  be  of  quite  large  diameter,  making 
the  speed  of  the  armature  slow.  The  opposite  poles  are 
in  this  case  of  the  same  sign,  and  alternate  poles  of  oppo- 
site sign.  An  inspection  of  Figs.  16  and  17  shows  that 
ring  armatures  must  require  more  wire  than  drum  arma- 
tures for  a  given  output,  other  things  being  equal,  and 
therefore  they  must  have  greater  self-inductance.  The 
mechanical  arrangement  of  surface-wound  coils  is,  how- 


Fig.  17 


ever,  more  secure  on  the  ring,  but  the  advantages  of 
lathe  winding  cannot  be  secured.  The  possibility  of  a 
comparatively  slow  speed  of  revolution  which  is  pos- 
sessed by  the  ring  has  commended  it  to  some  Euro- 
pean and  English  builders  of  dynamos,  but  the  toothed 
armature  for  alternators  has  not  been  as  fully  developed, 
nor  is  it  as  well  thought  of,  there  as  in  America. 

10.  Disc  Armatures.  —  The  disc  form  of  armature  for 
alternators  is  the  earliest  that  came  into  service.  The 
first  commercial  alternator  was  a  magneto-machine  (i.e. 
with  permanent  field  magnets)  known  as  the  Alliance 


ARMATURE  WINDINGS  FOR  ALTERNATORS.       2/ 

dynamo.  This  was  originally  devised  as  early  as  1849, 
but  was  not  developed  into  commercial  form  until  after 
1860.  In  1867  Wilde  built  an  alternator  with  electro- 
magnets and  a  disc  armature.  Since  that  time  the  disc 
armature  has  received  much  attention  in  Europe  and 
England,  and  has  been  an  element  of  many  successful 
machines.  (Examples  :  Siemens,  Ferranti,  and  Mordey 
alternators.)  The  disc  has  received  less  attention  in 
America  than  it  deserves,  and  is  here  represented  by 
but  few  machines  that  are  in  commercial  service.  This 
may  be  partially  due  to  the  essential  peculiarity  of  the 
disc,  which  admits  of  no  iron  core,  and  is  therefore  diffi- 
cult to  build  in  a  substantial  and  workmanlike  manner. 
That  this  difficulty  can  be  overcome  is  shown  by  the 
success  of  foreign  alternators  with  disc  armatures.  Disc 
armatures  may  be  wound  either  with  coils  or  with  con- 
tinuous windings,  as  shown  in  Fig.  18.  Either  the 
armature  or  the  field  may  revolve.  (Examples  :  Ferranti 
alternator,  Mordey  alternator,  Brush  alternator.)  The 
absence  of  iron  in  disc  armatures  reduces  the  losses  due 
to  hysteresis  and  foucault  currents  to  a  minimum,  but 
it  is  likely  to  require  an  increase  in  the  depth  of  the 
air  gap.  Hence  a  greater  magnetizing  current  is  likely 
to  be  required,  or  many  turns  must  be  placed  upon  the 
armature.  That  this  difficulty  also  can  be  overcome  is 
shown  by  the  small  amount  of  energy  required  to  mag- 
netize the  Mordey  alternators.  In  a  75-kilowatt  ma- 
chine of  this  type,  the  C2R  loss  in  the  armature  is  2.3 
per  cent  and  in  the  field  1.5  per  cent,  which  compares 
favorably  with  continuous-current  machines  (see  Vol.  I., 
pp.  108  and  138).  The  curve  of  electric  pressure  in 


28 


ALTERNATING   CURRENTS. 


disc  armatures  is,  in  general,  quite  near  to  that  of  a 
sine  function.  The  first  experimental  determination  of 
the  form  of  the  curve  was  made  in  1880  by  Joubert, 
who  experimented  upon  a  Siemens  machine  having  a 
disc  armature.  The  curve  proved  to  be  practically  a 
sinusoid.  This  is  also  true  of  the  curve  of  pressure 
developed  by  a  Mordey  alternator.  In  iron-cored  ma- 


y 

1 

= 

r 

V 

N 

8 

— 

= 

Fig.  18  a 

chines,  armature  reactions  have  a  more  apparent  dis- 
torting effect  and  therefore  modify  the  form  of  the 
curve  (sometimes  to  a  considerable  degree).  The  vari- 
ation is  usually  not  great  in  surface-wound  machines, 
but  when  imbedded  armature  conductors  are  used,  the 
curves  deviate  widely  from  a  sinusoid  and  sometimes 
become  quite  irregular. 

11.    Pole   Armatures.  —  As  already  stated   (Sect.  8), 
there  is,  in  general,  no  distinct  division  between  pole 


ARMATURE  WINDINGS  FOR  ALTERNATORS.       29 

armatures  and  drum  armatures  with  imbedded  con- 
ductors. In  order  to  avoid  foucault  currents  in  the 
pole  pieces,  and  sharp  corners  in  the  curves  of  arma- 
ture pressure,  precautions  must  be  taken  to  make  the 
magnetic  surface  of  toothed  armatures  uniform,  exactly 
as  is  done  in  the  case  of  toothed  continuous-current 


Pig.  18  b 

armature  cores  (Vol.  I.,  p.  154).  If  this  is  not  done, 
it  is  necessary  to  thoroughly  laminate  the  field  mag- 
nets, since  the  fluctuations  of  the  induction  in  the 
pole  pieces,  and  sometimes  in  the  whole  magnetic  cir- 
cuit, are  likely  to  become  very  much  greater  and  more 
rapid  than  ever  occurs  in  continuous-current  machines. 
(Example  :  Ganz  alternator.)  This  ordinarily  entails  an 
excessive  expense,  but  in  the  Ganz  alternator  the  use 
of  segmental  punchings  for  both  field  and  armature 


30  ALTERNATING   CURRENTS. 

(Fig.  19)  probably  reduces  the  extra  cost  of  the  con- 
struction to  some  extent.  The  effect  of  a  uniform 
reluctance  in  the  magnetic  circuit  may  be  gained  by 
placing  narrow,  deep  armature  coils  in  grooves  as 
shown  in  Fig.  20.  (Example :  Giilcher  alternator.) 
This  is  similar  in  many  respects  to  the  earliest  alter- 


ARMATURE 


Pig.  19 


nator  in  which  the  pole  type  of  armature  was  used, 
which  was  built  by  Lontin  without  provision  being 
made  to  secure  an  unvarying  reluctance.  If  the  pole 
pieces  of  both  field  and  armature  are  made  wide  and 
quite  close  together,  the  changes  in  the  reluctance  of 
the  magnetic  circuit  probably  may  be  reduced  so  as 
to  be  practically  negligible  (example :  Hopkinson  alter- 
nator), but  armature  reactions  and  magnetic  leakage 


ARMATURE  WINDINGS  FOR  ALTERNATORS.       31 

must  be  much  increased  by  this  construction  (Fig.  21). 
In  alternators  having  pole  armatures,  it  is  evident  that 
either  the  fields  or  the  armatures  may  be  arranged  to 
revolve. 

12.    Collectors.  —  In  alternators  having  either  type  of 
armature  here  described,  a  portion  of  the  magnetic  cir- 


ARMATURE 


Fig.  20 

cuit  which  carries  either  the  field  or  the  armature  wind- 
ings must  be  arranged  to  revolve.  In  some  cases  where 
the  field  magnets  compose  the  revolving  part  it  is  pos- 
sible to  arrange  the  construction  so  that  the  magnetizing 
coils  may  remain  stationary,  but  mechanical  considera- 
tions usually  render  this  inadvisable.  It  may,  therefore, 
be  said  generally  that  either  the  armature  or  field  wind- 
ings must  revolve  with  the  iron  on  which  they  are  wound. 


32  ALTERNATING   CURRENTS. 

This  makes  essential  the  use  of  some  means  for  convey- 
ing the  current  to  and  from  the  revolving  coils.  In  either 
case,  no  commutation  is  required,  and  therefore  plain, 
insulated  Collecting  Rings  serve  the  purpose  when  they 
are  properly  mounted  on  the  shaft  so  that  brushes  may 
be  arranged  to  bear  against  them.  These  rings,  often 
called  Collectors,  are  usually  made  of  copper  or  bronze. 


Fig.  21 

Instead  of  brushes,  the  collection  may  be  effected  by 
means  of  flexible,  weighted  copper  bands  hung  over  the 
rings  or  by  similar  arrangements  (Fig.  22).  By  the  lat- 
ter construction  a  large  collecting  area  may  be  gained 
without  unduly  increasing  the  width  of  the  rings.  The 
current  collected,  per  square  inch  of  collecting  contact, 
usually  varies  between  100  and  500  amperes.  The  safe 
limit  is  fixed  by  heating  alone,  and  therefore  may  be 
quite  large  without  danger.  If  mechanical  considera- 
tions required  it,  doubtless  more  than  500  amperes  per 


ARMATURE  WINDINGS  FOR  ALTERNATORS.       33 

square  inch  of  contact  might  be  satisfactorily  collected, 
but  as  such  extreme  cases  do  not  often  arise,  it  is  not 
generally  advisable  to  exceed  200  amperes  per  square 
inch. 


Fig.  22 

13.  Inductor  Alternators.  —  The  windings  of  inductor 
alternators  may  be  made  entirely  stationary,  thus  avoid- 
ing collecting  devices.  (Examples :  Stanley  and  War- 
ren alternators.)  These  devices,  however,  are  of  so 
little  expense  in  construction  and  material  that  there 
is  no  marked  advantage  in  suppressing  them,  and  the 
mechanical  and  magnetic  difficulties  encountered  in  the 
design  and  construction  of  inductor  alternators  has  not 
D 


34 


ALTERNATING   CURRENTS. 


permitted  them  to  come  into  general  use,  though  there 
are  several  theoretical  points  of  advantage  presented  in 
their  design.  In  order  to  avoid  excessive  losses  due  to 
foucault  currents  and  hysteresis  it  is  important  that  the 
induction  in  the  field  magnets  be  kept  as  uniform  as 

possible.  Since  this  can- 
not be  fully  accomplished,  it 
is  necessary  to  thoroughly 
laminate  the  iron  in  which 
the  induction  varies.  The 
inductor  must  be  moved  in 
such  a  way  as  to  periodi- 
cally short-circuit  or  break 
the  lines  of  force  which 
naturally  pass  through  the 
armature  coils.  This  may 
be  accomplished  as  shown 
in  Fig.  23,  where  the  effec- 
tive reluctance  of  the  total 
magnetic  circuit  is  fairly 
constant  for  all  positions  of 
the  inductor.  (Example  : 
Kingdon  alternator.)  The 
figure  shows  that  the  re- 
luctance cannot  remain  en- 
tirely constant,  and  that 
the  effective  ampere  turns 
Fig-  23  in  the  magnetic  circuits 

also  vary  with  the  position  of  the  inductor.  In  Figs. 
24,  25,  and  26  are  shown  two  types  of  inductor  machines 
in  which  no  attempt  is  made  to  keep  the  magnetic  cir- 


ARMATURE  WINDINGS  FOR  ALTERNATORS.       35 


cuit  of  constant  reluctance,  but  in  the  latter  the  iron  in 
the  fields  has  been  reduced  to  the  minimum  bulk.  Each 
of  these  forms  gains  some  economy  in  construction  by 
uniting  the  coils.  (Examples  :  Stanley,  Royal,  and  War- 
ren alternators.) 


Fig.  24 

In  the  Stanley  and  Warren  alternators  lines  of  force 
are  caused  by  the  motion  of  the  inductor  to  sweep 
across  the  armature  coils,  while  the  total  magnetism  in 
the  inductor  remains  fairly  constant.  The  field  wind- 
ings embrace  the  inductor  core,  and  the  machines  are 
not  true  inductor  alternators  but  would  be  properly 
classified  as  machines  having  drum  armatures  and  field 
magnets  of  a  modified  Mordey  type. 

14.  Armature  Insulating  and  Core  Materials.  — Before 
the  conductors  are  placed  on  an  armature  core  it  is 


ALTERNATING   CURRENTS. 


usual  to  insulate  it,  very  much  as  in  the  case  of  a 
continuous-current  armature  (Vol.  I.,  p.  104),  but  more 
thoroughly  on  account  of  the  high  pressures  usually 
produced  in  alternator  armatures.  For  this  purpose 
mica,  micanite,  mica  cloth,  shellacked  canvas,  fuller 
board,  oiled  paper,  sheets  of  vulcanite,  vulcabeston, 
vulcanized  fibre,  and  similar  insulating  materials  are 


Fig.  25 


used.  Mica,  micanite,  vulcanite,  vulcabeston,  bonsilate, 
vulcanized  fibre,  asbestos  paper,  and  similar  materials 
are  also  used  to  insulate  collecting  rings  and  brush 
holders,  and  for  insulation  between  the  armature  coils. 
The  wire  used  for  high-pressure  alternator  armatures  is 
often  triple-cotton  covered,  and  is  thoroughly  japanned 
during  the  process  of  winding.  Vulcanized  fibre  is  made 
from  paper  fibre  by  a  chemical  process  and  is  furnished 


ARMATURE  WINDINGS  FOR  ALTERNATORS.       37 

in  sheets  and  tubes.  Its  convenient  form,  cheapness, 
and  ease  of  working  have  brought  it  into  extensive  use. 
It  unfortunately  absorbs  moisture  when  exposed  to  the 
air,  which  causes  it  to  expand  and  contract  to  a  remark- 
able degree.  The  moisture  also  reduces  its  insulating 
qualities  to  a  large  degree.  It  is  therefore  unsafe  to 
place  entire  reliance  upon  it  where  continuously  high 
insulation  is  required.  Of  the  various  available  insulat- 
ing materials,  mica  is  the  only  thoroughly  reliable  one, 


Fig.  26 

but  it  is  unduly  expensive  and  of  poor  mechanical  qual- 
ities. On  the  latter  account  it  is  generally  used  in  com- 
bination with  other  materials.  When  made  up  in  the 
form  of  micanite  by  treating  with  varnish  and  subjecting 
to  high  pressure  and  heat,  its  mechanical  qualities  are 
somewhat  improved,  and  it  may  be  formed  into  sheets 
and  tubes  or  moulded  as  desired.*  Materials  such  as 
vulcabeston  and  bonsilate  are  advantageous  for  insulat- 
ing collector  rings  and  similar  details,  since  they  can  be 
moulded  into  any  desired  form.  Vulcabeston,  which  is  a 

*  Trans.  Amer.  Inst.  E.  £.,  Vol.  9,  p.  798. 


38  ALTERNATING   CURRENTS. 

compound  containing  rubber  and  asbestos,  is  also  manu- 
factured in  sheets  and  has  quite  satisfactory  mechanical 
and  electrical  properties.  Bonsilate  is  not  sufficiently 
tough  for  very  general  service.  Boxwood  and  paraffined 
maple  or  mahogany  are  frequently  used  for  insulation 
where  considerable  bulk  is  required  and  their  mechani- 
cal properties  will  serve,  though,  as  they  are  liable  to  be 
more  or  less  affected  by  the  surrounding  condition  of 
humidity,  they  must  be  used  with  care. 

The   armature    cores    themselves   may   be   made   of 
punched  iron  discs,  iron   punchings  of   special  shapes, 
iron  wire,  or  iron  tape.     In  American  machines  the  use 
of^discs  is  almost  universal.     (Examples  :  General  Elec- 
tric    and    Westinghouse 
machines.)       The    Kapp 
machines  have  armature 
cores  made  of  iron  tape.* 
Special     punchings     are 
used   in   machines    made 
Fifir>  27  by  Ganz  &  Co.  (Fig.  19), 

Siemens  &  Halske,  Giilcher,  Elwell  &  Parker,  and  other 
foreign  manufacturers,  although  foreign  manufacturers 
also  use  punched  discs  to  some  extent.  In  the  larger 
sizes  of  American  machines,  which  are  built  for  very 
low  speeds  to  connect  directly  to  steam  engines,  and 
therefore  have  armatures  of  large  diameters,  the  discs 
are  usually  built  up  out  of  segmental  punchings  put 
together  in  such  a  way  that  the  segments  of  alternate 
layers  break  joints  (Fig.  27). 

*  Kapp's  Dynamos,  Alternators,  and  Transformers,  p.  467. 


SELF-INDUCTION   AND   CAPACITY.  39 


CHAPTER   III. 

SELF-INDUCTION    AND    CAPACITY. 

15.  Self-induction.  —  Before  proceeding  with  the 
development  of  alternator  design,  it  is  essential  to 
discuss  the  relation  which  exists  between  electric 
pressures  and  currents  in  a  circuit  which  carries  an 
alternating  current.  It  is  to  be  remembered  that  dhe 
current  produced  by  cutting  lines  of  force  sets  up  in 
turn  magnetic  lines,  which  are  in  opposition  to  the 
original  field.  Again,  when  a  current  is  introduced 
into  a  circuit,  it  produces  a  magnetic  field  the  rise  of 
which  causes  a  counter  electric  pressure.  These  condi- 
tions are  necessary  under  the  law  of  conservation  of 
energy  and  its'  corollary,  Lenz's  Law  (see  Vol.  I.,  p.  77). 
This  counter  electric  pressure  is  called  the  Electric 
Pressure  or  Electromotive  Force  of  Self-induction,  and 
the  phenomenon  as  a  whole  is  called  Self-induction. 
Therefore  self-induction  may  be  defined  as  the  inherent 
quality  of  electric  currents  which  tends  to  impede  the 
introduction,  variation,  or  extinction  of  an  electric  cur- 
rent passing  through  an  electric  circuit.  The  electric 
pressure  in  a  circuit  which  is  due  at  any  instant  to  self- 
induction  is  evidently  proportional  to  the  rate  of  change 
of  the  magnetization  set  up  by  the  current  of  the  circuit ; 
therefore,  magnetic  reluctance  remaining  constant,  the 


4o 


ALTERNATING   CURRENTS. 


electric  pressure  of  self-induction  is  proportional  to  the 
rate  of  change  of  current  in  the  circuit,  which  makes  it 
again  proportional  to  the  rate  of  change  of  instanta- 
neous pressure  producing  the  current.  The  pressure 
which  is  effective  in  producing  current  may  be  called 
the  Active  Pressure,  to  distinguish  it  from  the  pressure 
which  is  Impressed  upon  the  circuit.  The  latter  is 
called  the  Impressed  Pressure.  It  is  evident  that  the 
active  pressure  at  any  instant  is  equal  to  the  difference 
between  corresponding  instantaneous  values  of  the  im- 
pressed pressure  and  of  the  counter  pressure.  The 


corresponding  instantaneous  current  in  the  circuit  is 

given  by  Ohm's  law,  c  =  — ,  where  c  and  e  are  instan- 

R 

taneous  current  and  pressure  in  amperes  and  volts,  and 
R  is  the  resistance  of  the  circuit  in  ohms.  A  little 
consideration  will  show  that  the  Phase  of  the  electric 
pressure  of  self-induction  is  not  in  unison  with  that  of 
the  current,  although  their  periods  are  the  same ;  be- 
cause, the  counter  pressure  is  proportional  to  the  rate  of 
change  of  magnetism  caused  by  the  current,  and,  sup- 
posing the  current  to  be  an  alternating  one,  its  rate  of 


SELF-INDUCTION   AND   CAPACITY.  41 

change  is  zero  when  it  is  a  maximum,  and  the  pressure 
of  self-induction  is  therefore  zero  at  the  same  time. 
When  the  curve  of  current  is  a  sinusoid,  the  rate  of 
change  of  its  ordinates  at  any  point  is  proportional  to 
— (  in  a)  _  cos  a  a  __  _  sm(a  —  go  )  a  p^ence  ^Q  curve 

dt  dt  dt 

of  counter  electric  pressure  is  a  sinusoid,  the  phase  of 

which  lags  90°  behind  the  phase  of  the  current,  and 
therefore  of  the  active  pressure.  Suppose  the  line 


OC  (Fig.  28)  represents  the  maximum  value  of  the 
active  pressure.  If  the  line  be  uniformly  rotated 
around  the  point  O,  its  end  C  describes  a  circle  and 
its  vertical  projection  a  simple  harmonic  motion.  At 
each  instant  the  vertical  projection  of  the  line  pro- 
portionally represents  the  magnitude  of  the  instanta- 
neous *  active  pressure  corresponding  to  the  angular 
advance  of  the  line.  The  projection  of  the  line  OD, 
which  is  90°  behind  OC,  likewise  represents  the  instan- 


42  ALTERNATING   CURRENTS. 

taneous  counter  electric  pressure  at  each  instant.  The 
algebraic  sums  of  the  instantaneous  projections  of  OC 
and  OD,  are  always  equal  to  the  simultaneous  projections 
of  OA.  But  OC2  =  OA*  -  OD2  or  OC  =  -VOA*  -  OD*. 
The  lengths  of  the  lines  have  been  assumed  to  be 
proportional  to  maximum  values  of  pressures,  but  the 
effective  values  of  the  pressures  hold  the  same  relation 
since  each  effective  value  is  equal  to  the  maximum 
multiplied  by  .707.  Consequently,  the  active  pressure 
operating  in  a  circuit  with  self-induction,  is  equal  to  the 
square  root  of  the  difference  between  the  squares  of 
the  impressed  and  the  inductive  electric  pressures.  Or 
when  Ea,  E{,  and  E.  are  respectively  the  active,  im- 
pressed, and  inductive  effective  pressures, 


The  angle  $  between  the  lines  OA  and  OC  shows  the 
amount  by  which  the  phase  of  the  active  pressure  lags 
behind  that  of  the  impressed  pressure.  The  figure 
makes  it  evident  that  this  lag  is  caused  by  the  posi- 
tion of  the  self-induction  line,  and  that  its  magnitude 
depends  upon  the  length  of  that  line.  The  tangent 


of  the  angle  is  tan  0  =         =  Inductive  Pressure 

OC       Active  Pressure 

called  the   Angle   of   Lag.     The   relations   are   plainly 
shown  in  Fig.  2ga. 

16.  Self  -Inductance.  —  The  pressure  of  self-induction 
is  proportional  to  the  energy  exerted  by  the  current  in 
a  coil  while  setting  up  the  lines  of  force  which  surround 
it  (Vol.  I.,  p.  69).  Its  magnitude  is  therefore  dependent 
upon  the  number  of  lines  of  force  enclosed  by  the  circuit 


SELF-INDUCTION   AND   CAPACITY.  43 

per  unit  current,  the  number  of  turns  composing  the  cir- 
cuit, and  the  current  flowing  therein.  From  the  reference 

just  given,  E  —  —Q  —  ,  where  E  is  the  pressure  of  self- 
icrdt 

induction  for  the  given  rate  of  change  of  magnetiza- 
tion, n  is  the  number  of  turns  in  the  coil,  and,  as 
before,  N  is  the  magnetic  flux.  But  in  a  long  solenoid, 

IV—  ^'JTn  _  __,  where  n1  is  the  number  of  turns  per  cen- 

10 
timeter,  and  A  is  the  area  of  the  solenoid.     Then 


10  io9          dt 

4irnn'A  is  called  the  absolute  Self-Inductance  or  the 
Coefficient  of  Self-induction  of  the  coil,  and  is  usually 
represented  by  the  capital  letter  L.  ^.Trn'A  is  equal 
to  the  number  of  lines  of  force  passing  through  the 
solenoid  when  the  current  is  one  C.G.S.  unit.  Hence 
the  C.G.S.  value  of  the  self-inductance  of  any  circuit 
which  is  in  a  magnetic  medium  of  unit  permeability, 
may  be  defined  as  the  product  of  the  number  of  lines 
of  force  enclosed  by  the  circuit  when  carrying  a  unit 
current,  by  the  number  of  turns  in  the  circuit,  or 
La  —  nNv  Since  the  number  of  lines  of  force  devel- 
oped by  a  coil  is  proportional  to  the  number  of  turns 
composing  it,  its  self-inductance  is  proportional  to 
the  square  of  its  turns.  In  order  that  the  formula 


x  be  wrinen  E  =  the    val. 

io9  dt  dt 

ues   of    C  and  E  being   given  in  amperes  and  volts, 

the  practical  value  of  L  is  made   io9  times  as  large  as 
that  of  the  absolute  unit.     The  Chicago  Electrical  Con- 


44  ALTERNATING   CURRENTS. 

gress  has  formally  declared  the  name  of  this  practical 
value  to  be  the  Henry.  The  formula  plainly  shows 
that  the  absolute  dimension  of  the  C.G.S.  unit  of  self- 
inductance  is  one  centimeter,  and  the  practical  unit  is 
therefore  io9  centimeters  or  one  theoretical  earth  quad- 
rant. The  name  Quadrant  was  therefore  at  one  time 
assigned  to  the  henry.  Since  the  absolute  dimensions 
of  the  ohm  are  that  of  a  velocity  equal  to  one  earth 
quadrant  per  second,*  the  dimensions  of  the  henry  are 
equal  to  the  ohm  multiplied  by  one  second,  and  there- 
fore the  term  Secohm  (second-ohm),  a  name  suggested 
by  Ayrton  and  Perry  for  the  unit,  came  into  some  use. 
The  use  of  the  name  henry,  in  honor  of  Professor 
Joseph  Henry  of  Princeton  College,  was  suggested  by 
the  American  Institute  of  Electrical  Engineers,  and 
was  officially  adopted  by  the  International  Electrical 
Congress  at  Chicago. f  The  name  henry  is  therefore 
now  the  proper  and  only  name  for  the  unit. 

The  definition  of  the  henry  is  developed  for  a  long 
solenoid  in  a  medium  of  unit  permeability.  If  the  cir- 
cuit does  not  comprise  a  long  solenoid,  the  general  defi- 
nition still  holds,  as  already  said,  but  the  summation  of 
the  number  of  lines  of  force  passing  through  each  turn 
individually  must  be  taken,  since  the  number  of  lines 
passing  through  the  turns  is  a  variable  which  depends 
upon  their  position  in  the  coil.  Thus,  suppose  Fig. 
30  to  represent  a  short  solenoid  of  eight  turns  in  which 
are  developed  ten  lines  of  force  when  one  ampere  flows 
through  the  coil.  Assuming  the  distribution  of  the 

*  Thompson's  Elect,  and  Mag. ,  revised  edition,  Art.  357. 

t  Proceedings  International  Electrical  Congress  of  Chicago,  p.  1 8. 


SELF-INDUCTION   AND   CAPACITY. 


45 


lines  shown  in  the  figure,  the  inductance  is  calculated 
as  follows : 


10x24-8x2+6x2+4x2 

=  56  C.G.S.  units,  or  ~g  henrys. 

If    an    iron    core    be    now    placed    in    the    coil,    the 
number   of   lines   of   force    will    be   increased    directly 


Pig.  30 

as  the  reluctance  of  the  magnetic  circuit  is  de- 
creased. Hence,  assuming  the  distribution  of  the 

lines   to   remain    unchanged,   the   inductance   becomes 

56  P'  P' 

— »-=.  henrys,  where    -=-    is    the    ratio    of    the    reluc- 

icrP  P 

tance  before  and  after  the  iron  core  is  inserted.  In 
the  case  of  a  long  solenoid  L  =  --^^ — ,  when  the 
permeability  is  unity,  but  when  the  permeability  of 


46  ALTERNATING   CURRENTS. 

the  magnetic  circuit  taken   as  a  whole  is  /JL,  the  num- 

ber of  lines  of  force  due  to  unit  current  is  —  -, 

T      4  Trnn'Afi 
and  therefore  L  =  --  ^  --       In  general,   where  the 

magnetic  circuit  is  composed  wholly  of  non-magnetic 

material,  the  self-inductance  is  L  =  —  ^  —  =  n  0    ,    and 

iosC 


is  a  constant  for  all  values  of  C.  When  iron  or  other 
magnetic  material  is  included  in  the  magnetic  cir- 
cuit, the  value  of  the  self-inductance  varies  with  the 


value   of   C.     As   before,  L==~y  but    this 

io8C 


may  have  a  different  value  for  each  value  of  C,  since 

N     4  Trn'Au,  .          .  .     ~      T       .  . 

^  =  —          —  ,  and  u,  varies  with  C.     In  this  case  the 

C  10 

inductance  for  any  value  of  C  is  /JL  times  as  great  as 
when  no  magnetic  material  is  included  in  the  magnetic 
circuit,  the  value  of  //,  taken  being  that  corresponding 
to  the  particular  value  of  C.  The  self-inductance  of 
a  long  solenoid  which  contains  an  iron  core,  when  carry- 
ing a  certain  current,  may  therefore  be  defined  as  the 
number  of  turns  in  the  solenoid  multiplied  by  the 
number  of  lines  of  force  set  up  by  the  current  divided 
by  the  current.  As  an  example  of  the  calculation  of 
the  value  of  L,  consider  a  uniform  ring  of  wrought 
iron  100  centimeters  in  mean  circumference  and  20 
square  centimeters  in  cross-section.  Suppose  a  coil 
of  2500  turns  be  uniformly  wound  on  the  ring,  and  a 
current  of  two  amperes  be  passed  through  the  mag- 
netizing coil.  Taking  //,  as  equal  to  250,  which  is  a  fair 

value,  ^^-  =  47r  x  25  x  20  x  250=  1,571,000.      Hence 

2500  x  1,571,000  T,    . 

£  --  _j       -  i^-L       —  —  2.93  henrys.     If  the  current  in 


SELF-INDUCTION   AND   CAPACITY.  47 

the  magnetizing  coil  be  taken   as    ij,  the  value  of  fi 

2500  x  2,199,400 
becomes  roughly  350,  and  L  —  -  g  —       -=  5.50 

henrys.     If  the  ring  be  of  brass  or  other  non-magnetic 
material,  the  values  figure  out  as  follows  : 


=47r  x  25  x  20  =  6284, 


,      2500  x  6284 
and  L=—    -  g—  -=  .0157  henrys. 

In  the  usual  practical  problems  that  are  met,  the  con- 
formation and  numerical  constants  of  the  magnetic  cir- 
cuit and  its  windings  are  unknown,  or  are  so  irregular 
that  the  self-inductance  cannot  be  determined  by  calcula- 
tion, and  experimental  determination  must  be  resorted  to. 

At  the  meeting  of  the  Chicago  Electrical  Congress 
a  general  definition  was  given  for  the  henry  as  follows  : 
"As  a  unit  of  induction,  the  henry,  which  is  the  induc- 
tion in  a  circuit  when  the  electromotive  force  induced 
in  this  circuit  is  one  international  volt,  while  the  in- 
ducing current  varies  at  the  rate  of  one  ampere  per 
second."  This  is  in  agreement  with  the  definition 
already  presented. 

17.  Examples  of  Self  -Inductances.  —  Ordinary  practi- 
cal experience  in  electrical  measurements  and  in  hand- 
ling wires  soon  gives  a  capacity  for  estimating  the  value 
of  resistances  ;  in  the  same  way  facility  is  soon  gained 
in  roughly  estimating  electrostatic  capacities,  or  the 
current  which  may  be  safely  carried  by  a  wire,  or  even 
the  ampere-turns  required  to  produce  a  given  magnetiza- 
tion in  a  magnetic  circuit.  Ordinary  practice,  however, 
gives  little  clue  to  estimating  the  self-inductance  in  a 


48  ALTERNATING   CURRENTS. 

circuit.  It  is  true  that,  as  already  shown,  the  self- 
inductance  is  dependent  upon  the  magnetism  enclosed 
in  the  circuit  and  the  turns  thereof,  but  experience  in 
dealing  with  coils  and  magnetic  circuits  is  not  usually 
regarded  in  such  a  way  as  to  aid  in  estimating  self- 
inductances.  The  following  values  of  self-inductance 
are  therefore  presented  here  to  give  a  foundation  for 
judgment.* 

The  range  of  self-inductances  met  in  practice  is  very 
wide.  The  smallest  which  are  practically  met  are  in  the 
doubly  wound  resistance  coils  used  for  Wheatstone 
bridges  and  similar  devices.  Since  the  wire  in  these  is 
doubled  back  upon  itself,  the  magnetic  effect  of  the  cur- 
rent is  almost  neutral  and  the  inductance  is  often  less 
than  a  microhenry.  The  inductance  of  a  certain  electric 
call-bell  of  2.5  ohms  resistance  has  been  found  to  be  12 
microhenrys  ;  a  telephone  call-bell  of  80  ohms  resist- 
ance, 1.4  henrys;  the  armature  of  a  magneto  calling 
generator  of  550  ohms  resistance,  from  2.7  henrys  when 
the  plane  of  the  coil  lay  in  the  plane  of  the- pole  pieces, 
to  7.3  henrys  when  the  plane  of  the  coil  was  perpen- 
dicular to  the  plane  of  the  pole  pieces  ;  a  Bell  telephone 
receiver  measuring  75  ohms,  with  diaphragm,  75  to  100 
millihenrys,  without  diaphragm  about  35  per  cent  less; 
mirror  galvanometers  vary  with  their  resistance  from  a 
few  millihenrys  to  10  or  12  henrys;  a  mirror  galva- 
nometer for  submarine  signalling  of  2250  ohms  resist- 

*  Compare  Kennelly  on  Inductance,  Trans.  Atner.  Inst.  of  E.  E., 
Vol.  8,  p.  2;  Sumpner  on  Measurements  of  Inductance,  Jour.  Institution 
of  E.  E.,  Vol.  1 6,  p.  344;  Modern  American  Telegraphic  Apparatus,  Elec- 
trical Engineer,  Vol.  13,  etc. 


SELF-INDUCTION    AND   CAPACITY. 


49 


ance,  3.6  henrys ;  astatic  mirror  galvanometers  of  5000 
ohms  resistance  average  about  2  henrys.  The  single 
coil  of  a  Thomson  galvanometer  of  2700  ohms  resist- 
ance measured  2.56  henrys  ;  the  coil  of  another  Thom- 
son galvanometer  having  100,000  ohms  resistance 
measured  70  henrys ;  the  coil  of  an  Ayrton  and  Perry 
spring  voltmeter,  without  iron  core,  measured  1.462 
henrys.  This  coil  had  a  length  of  2.88  inches,  an  ex- 
ternal diameter  of  3  inches,  was  wound  on  a  brass  tube 
.58  inch  in  external  diameter,  and  had  a  resistance  of 
333-5  ohms.  Each  of  the  above  measurements  was 
made  with  a  current  of  a  few  milliamperes.  The  follow- 
ing are  measurements  of  telegraphic  apparatus  : 


POLARIZED   RELAYS   OF  VARIOUS  TYPES. 


Type. 

Resistance  in 
Ohms. 

Self-inductance  in 
Henrys. 

Testing  Current  in 
Milliamperes. 

I 

419 

1.99 

6-3 

2 

423 

1.89 

6-3 

3 

413 

1.69 

6-3 

4 

413 

I-3I 

6-3 

All  armatures  were  4  mils  from  poles. 

A  common  Morse  relay  of  148  ohms  resistance  meas- 
ured 10.47  henrys  with  the  armature  against  the  poles, 
and  3.71  henrys  with  the  armature  20  mils  from  the 
poles,  the  measuring  current  being  6.3  milliamperes. 
In  ordinary  working  adjustment,  the  inductance  of  a 
Morse  relay  is  about  5  henrys.  Telegraph  sounders  with 
bobbins,  respectively,  i^  by  i  and  1 1-  by  i^  inches,  each 


50  ALTERNATING   CURRENTS. 

wound  to  20  ohms  resistance,  measured  191  and  150 
millihenrys,  the  armatures  being  4  mils  from  the  poles 
and  the  measuring  current  being  125  milliamperes.  A 
single  coil  of  a  Morse  sounder  with  a  resistance  of 
32  ohms,  and  having  an  iron  core  .31  inch  in  diam- 
eter and  3  inches  long,  the  bobbin  being  .94  inch  in 
diameter,  was  found  to  have  a  self-inductance  of  94 
millihenrys.  A  complete  sounder  with  a  core  like  that 
of  the  preceding  coil,  but  with  a  bobbin  of  50  ohms 
resistance  having  a  diameter  of  1.25  inches,  was  found 
to  have  a  self-inductance  of  444  millihenrys.  The  self- 
inductance  of  a  complete  sounder  of  14  ohms  resistance 
measured  265  millihenrys. 

Bare  No.  12  B.  and  S.  gauge  copper  wire  erected 
on  a  pole  line  about  23  feet  from  the  ground,  is  calcu- 
lated by  Kennelly  to  measure  about  8.5  ohms  and  3.15 
millihenrys  per  mile  ;  number  6  copper  wire  under  simi- 
lar conditions  is  calculated  to  measure  about  2.1  ohms 
and  2.95  millihenrys.  A  quadruplex  telegraph  line, 
with  all  instruments  in  circuit,  measures  approximately 
10  henrys. 

The  largest  self-inductances  met  in  practice  are 
usually  in  the  windings  of  induction  coils  or  of  electrical 
machinery.  The  secondary  of  an  induction  coil  capable 
of  giving  a  two-inch  spark  and  having  a  resistance  of 
5700  ohms,  measured  51.2  henrys.  The  primary  of  an 
induction  coil  which  is  19  inches  long  and  8  inches  in 
diameter,  measured  .145  ohm  and  13  millihenrys,  while 
its  secondary  measured  30,600  ohms  and  2000  henrys. 
The  inductance  of  dynamo  fields  is  likely  to  vary  from 
I  to  1000  henrys  ;  continuous-current  dynamo  armatures 


SELF-INDUCTION   AND   CAPACITY.  51 

measure  between  the  brushes  from  .02  to  50  henrys ; 
the  fields  of  a  shunt-wound  Mather  and  Platt  continuous- 
current  dynamo  built  for  an  output  of  100  volts  and  35 
amperes,  measured  44  ohms  and  13.6  henrys  at  a  small 
excitation  ;  the  armature  of  the  same  machine  measured 
.215  ohm  and  .005  henry;  a  Mordey  alternator  armature 
of  the  disc  type,  with  a  capacity  for  18  amperes  at 
a  pressure  of  2000  volts,  measured  2  ohms  and  .035 
henry ;  a  Kapp  alternator  armature  of  the  ring  type, 
with  a  capacity  of  60  kilowatts  at  2000  volts  pressure, 
measured  1.94  ohms  and  .069  henry;  another  Kapp 
machine,  30  kilowatts  2000  volts,  measured  7  ohms  and 
.0977  henry  ;  the  fields  of  a  Ferranti  alternator  meas- 
ured 3  ohms  and  .61  henry,  while  the  armature  of  the 
same  machine  built  for  an  output  of  200  volts  and  40 
amperes  measured  .0011  to  .0013  henry,  with  no  cur- 
rent in  fields  ;  the  primary  and  secondary  windings  of 
transformers  measure  roughly  from  .001  of  a  henry  up 
to  50  henrys,  depending  upon  their  output  and  the 
pressure  for  which  they  are  designed. 

The  effect  of  the  field  magnetism  upon  the  self- 
inductance  of  a  disc-alternator  armature  is  shown  by 
some  measurements  taken  by  Dr.  Duncan*  on  a  small 
Siemens  eight-pole  alternator,  the  results  of  which  are 
given  in  the  table  on  the  following  page. 

Professor  Ayrton  found  that  the  self-inductance  of  an 
unexcited  Mordey  alternator  armature  varied  between 
.033  and  .038  henry,  and  that  this  decreased  about 
10  per  cent  when  the  fields  were  excited. f 

*  Electrical  World,  Vol.  II,  p.  212. 

t  Jour.  Institution  of  E.  E.,  Vol.  18,  p.  662,  also  ibid,  p.  654. 


52  ALTERNATING   CURRENTS. 

SELF-INDUCTANCE  OF  ARMATURE   IN   PLACE. 


Position  of  Armature. 

„,                    .       _-,.    -J 

0° 

nV3 

22^ 

o.    Amperes 

.120 

.112 

.IOO 

2.5         « 

.125 

•"5 

.108 

4-5 

.128 

•"5 

.IO6 

Self-induction  of  armature  removed  from  field,  .082 
henry;  resistance  of  armature,  .7  ohm;  pitch  of  the 
poles  45°. 

18.   The  Energy  of  the  Self-Induced  Magnetic  Field.  - 

It  has  been  shown  (Vol.  L,  p.  69)  that  the  work  done 
against  the  electric  current  when  the  number  of  lines  of 
force  passing  through  a  circuit  is  changed,  is 

-nCdN 


dW  = 


io8 


If  L  has  a  fixed  value,  then  CdN=  NdC,  and 
-  nNdC 


dW= 


io8 


=  -LCdC. 


Hence  the  work  stored  in  the  magnetic  field  when  the 
current  changes  from  a  zero  value  to  value  C  is 


W 


=  -  C°LCdC=- 

Jo 


LC* 


When  the  current  again  falls  to  zero,  work  is  restored 

LC2 
to  the  circuit  by  an  amount  equal  to  -  —  •     If  L  varies, 

as  is  the  case  where  magnetic  material  is  in  the  path  of 
the  lines  of  force,  CdN  is  no  longer  equal  to  NdC,  but 


SELF-INDUCTION   AND   CAPACITY.  53 


the  work  stored  in  the  field  still  remains  -  if  L   is 

given  its  average  value  between  the  limiting  values 
of  the  current.  If  there  is  hysteresis,  the  average 
value  of  L,  when  going  up  the  curve,  is  greater  than 
when  going  down  the  curve,  and  the  work  stored  in  the 
field  by  the  increasing  current  is  not  all  recovered  when 
the  current  falls  again  to  zero.  If  a  coil  be  wound  on  a 
closed  ring  of  soft  iron,  which  exhibits  great  retentive- 
ness  and  coercive  force  in  this  form,  the  value  of  L  is 
very  great  if  the  ring  be  magnetized  by  an  alternating 
current.  If  the  ring  be  magnetized  by  a  rectified  peri- 
odic current,  that  is,  one  which  varies  uniformly  between 
zero  and  a  maximum,  the  value  of  L  is  practically  the 
same  as  though  the  iron  core  were  not  present.  This 
behavior  is  due  to  the  ring  continuously  retaining  the 
magnetization  caused  by  the  maximum  current,  and 
since  the  induction  in  the  core  therefore  remains  con- 
stant it  does  not  set  up  a  counter  electric  pressure.  By 
making  a  cut  in  the  ring,  its  coercive  force  may  be  re- 
duced so  much  that  the  average  value  of  L  is  practically 
the  same  for  rectified  and  alternating  currents. 

19.  Curves  of  Rising  and  Falling  Currents  in  a  Self- 
Inductive  Circuit.  —  The  work  done  by  the  effect  of  self- 
inductance  is  manifested,  as  already  explained,  by  a 
counter  electric  pressure  which  tends  to  retard  a  rising 
current  and  to  accelerate  or  continue  a  falling  current. 
That  is,  it  produces  an  effect  in  many  respects  analo- 
gous to  the  inertia  of  tangible  matter.  In  the  case  of 


the  latter,  MV,  and        -      are   respectively  the 

2  dt 

energy,  momentum,  and  rate  of  change  of  momentum 


54 


ALTERNATING    CURRENTS. 


of  the  mass  Mt  when  moving  at  a  velocity  V\  while  in 

7"  /~"2  7"  x-//~* 

the  case  of  the  electric  circuit, ,  LC,  and  ,  may 

be  called  the  energy,  momentum,  and  rate  of  change  of 
momentum  (counter  electric  pressure)  of  its  magnetic 
field.  If  a  circuit  having  self-inductance  be  suddenly 
connected  to  a  source  of  constant  electric  pressure, 

E" 

the  current  does  not  rise  instantly  to  the  value  C=  — , 

R 

but  it  is  retarded  so  that  its  rise  is  always  along  a  loga- 
rithmic curve  (Fig.  31).  When  the  current  has  reached 


its  full  value,  a  smaller  quantity  of  electricity  has  passed 
through  the  circuit  during  the  interval  than  would  have 
passed  if  the  retardation,  or  momentum  effect,  had  not 
been  present.  This  decreased  amount  of  electricity  is 
proportional  to  the  area  OYQ  between  the  curve  of 
current  and  the  horizontal  line  YQ  (Fig.  31).  The 
counter  electric  pressure  at  any  instant  during  the  rise 
of  the  current  is  ^-  —  ,  and  the  instantaneous  current 


which  would  flow  through  the  circuit   under  its   influ- 
ence is  c  =  ^^  —  .     The  total  quantity  of  electricity 


SELF-INDUCTION    AND   CAPACITY.  55 

which  would  be  transferred  through  the  circuit  due  to 
a  change  of   the   induction  from    o  to  N  is  therefore 


=_.     This  is  equa!  to 
R 

the  deficit  of  electricity  which  flows  through  the  cir- 

cuit in  the  period  during  which  the  current  is  rising 

to  its  permanent  value  C  =  —.     If  the  pressure  be  sud- 

R 

denly  reduced  to  zero,  the  current  does  not  stop  imme- 
diately, but  falls  off  along  a  logarithmic  curve,  and  the 
quantity  of  electricity  passing  through  the  circuit  is 
increased  on  this  account.  The  increased  quantity  is 
proportional  to  the  area  between  the  curve  and  the  X 
axis.  That  this  quantity  is  equal  to  the  quantity  of 
electricity  lost  in  starting  the  current  is  shown  thus  : 
The  counter  electric  pressure  is,  as  before,  -  —  —  ,  and 

-ndN     ™  f°  C-dN      nN       LC 

Whence  I  cdt  —  n  I 
J*  &* 


c  = 


B  _ 
icPR       icPR      R 

This  is  equal  and  opposite  in  sign  to  the  quantity  lost 
upon  starting  the  current.  Hence,  if  the  induction 
passing  through  the  circuit  returns  to  its  initial  value, 
exactly  the  same  total  number  of  coulombs  passes 
through  a  circuit  having  self-inductance  as  would  pass 
were  there  no  self-inductance.  In  the  same  way  if 
a  mass  of  moving  matter  be  raised  from  a  velocity 
V  to  a  velocity  V,  a  certain  amount  of  work  is  done 
in  accelerating  the  body;  but  if  after  a  certain  dis- 
tance has  been  traversed  the  velocity  be  allowed  to 
fall  to  V  again,  the  work  of  acceleration  is  returned 
and  the  total  work  during  the  cycle  is  exactly  the 
same  as  if  inertia  did  not  exist  in  the  mass.  If  the 
electric  circuit  have  an  iron  core,  the  value  of  the  in- 


56  ALTERNATING   CURRENTS. 

duction  may  not  fall  to  its  initial  value  upon  breaking 
the  circuit,  and  the  energy  given  up  is  then  not  equal 
to  that  absorbed  in  building  up  the  magnetization. 
The  difference  in  the  energy  remains  stored  in  the 
magnetic  field  in  the  form  of  residual  magnetism.  As 
an  analogue,  suppose  that  when  the  moving  mass  as- 
sumed above  falls  in  velocity  it  comes  to  a  velocity 
V",  which  is  greater  than  the  initial  velocity  V.  Then 
some  of  the  energy  expended  in  acceleration  is  re- 
tained and  the  summation  of  the  work  clone  during 
the  cycle  is  increased  on  account  of  the  inertia  of 
the  body. 

The  condition  under  which  the  curves  of  rising  and 
falling  current  are  logarithmic  and  of  exactly  the  same 
dimensions  when  pressure  is  applied  and  withdrawn,  in- 
stantaneously, from  a  circuit  requires  that  the  resistance 
of  the  circuit  remain  constant.  If  the  circuit  is  broken, 
by  opening  a  switch  or  otherwise,  it  is  an  experimental 
fact  that  the  counter  pressure  rises  much  higher  than 
the  original  impressed  pressure,  frequently  rising  to 
many  times  its  value.  The  extreme  severity  of  the 
shock  which  may  be  received  upon  breaking  a  circuit  of 
large  inductance  attests  the  fact.  This  is  due  to  the 
exceedingly  large  increase  of  resistance  in  the  circuit, 
introduced  by  the  break.  The  increase  in  resistance 
causes  the  current  to  fall  off  more  quickly,  and  hence 
a  greater  rate  of  change  of  magnetism.  However,  as 
before,  the  work  given  out  by  the  field  must  be 

r  C'2 
—  , 

2 

and  is  equal  to  the  energy  stored  in  the  circuit  when 


r 
=  I 

Jo 


SELF-INDUCTION   AND   CAPACITY.  57 


/*<         Z£7 

the  current  was  introduced  ;  and  q  —  \cdt  =  —  —  ,  which 

is  the  same  as  before.  The  total  number  of  coulombs 
transferred  being  the  same  and  the  induced  pressure 
being  greater  upon  breaking  a  circuit  than  upon  making 
it,  the  period  of  action,  /,  must  be  shorter  upon  the 
break. 

20.  The  Effect  of  Self-Inductance  in  Divided  Circuits. 
Application  to  a  Shunted  Ballistic  Galvanometer.  —  The 
fact  that  the  total  quantity  of  electricity  which  passes 
through  a  wire  when  subjected  to  a  transient  electric 
pressure  is  independent  of  the  self-inductance  of  the  cir- 
cuit, as  is  shown  above,  has  a  bearing  upon  the  distribu- 
tion of  current  in  divided  circuits.  With  no  external 
disturbing  factors,  it  is  apparent  that  where  a  transient 
electric  pressure  is  impressed  upon  parallel  circuits  of  dif- 
ferent inductances,  the  number  of  coulombs  which  flow 
through  each  circuit  would  also  flow  were  the  circuits 
without  self-inductance,  but  the  phase  of  the  flow  in  each 
circuit  is  retarded  so  as  to  lag  behind  that  of  the  press- 
ure by  an  amount  which  is  proportional  to  the  self-induc- 
tance of  the  circuit.  This  reasoning  would  make  it  appear 
that  shunting  a  ballistic  galvanometer  must  change  the 
constant  in  the  ratio  of  the  resistances  of  galvanometer 
and  shunt  without  regard  to  their  self-inductances,  as  is 
true  when  continuous  currents  are  in  question.  This, 
however,  is  not  correct,  because  the  movement  of  the 
needle.  which  occurs  before  the  end  of  the  discharge 
generates  a  counter  electric  pressure  in  the  galvanome- 
ter coils.  This  reduces  the  proportion  of  the  discharge 
which  passes  through  the  galvanometer.  Assuming 


58  ALTERNATING   CURRENTS. 

that  the  number  of  lines  of  force  due  to  the  needle 
which  cut  the  coils  are  proportional  to  the  sine  of  the 
deflection  ;  calling  rg  and  L  the  resistance  and  self-in- 
ductance of  the  galvanometer  coils  ;  rg  the  resistance  of 
the  shunt  (the  inductance  of  the  latter  being  assumed 
negligible  on  account  of  its  being  wound  with  doubled 
wire)  ;  and  ca  and  cs  being  the  respective  instantaneous 
currents  ;  then  the  instantaneous  impressed  electric 
pressure  is  e  =  csrs.  The  instantaneous  active  pressure 
causing  currents  to  flow  through  the  galvanometer  coils 
is  cgrgt  and  is  equal  to  the  impressed  pressure  less  the 
counter  electric  pressure  caused  by  self-induction  and 
the  swing  of  the  needle.  Therefore 

(Ldca      /£</(sin  a) 


where  k  is  a  constant  which  depends  on  the  strength 
of  the  needle.     Whence  rscsdt  —  rgcgdt  =  Ldcg  +  kd(sm  a) 


and 


s(  csdt  —  rffJQ  cgdt  =  Lj^  dcg  +  £jo  </(sin  a). 


This  is  qsrs  —  qgrg  =  k  sin  a.  If  the  deflection  be  small 
then  sin  a  is  sensibly  equal  to  2  sin-,  but  K  sin  -  =  qg, 
where  K  is  the  ordinary  constant  of  the  ballistic  gal- 
vanometer (Vol.  I.,  p.  46).  Hence  qsrs  —  qgrg  =  —^ 
Calling  <2  the  total  discharge,  which  is  equal  to  qs  4-  gg, 
this  becomes 


r.(Q  ~  9.)  -  W»  = 


2k 


SELF-INDUCTION   AND   CAPACITY.  59 

This  discussion  shows  that  the  coulombs  of  a  discharge 
which  pass  through  a  shunted  ballastic  galvanometer 

are  less  than  the  ratio  of  the  resistances  or  qa  <        s   . 

29      r  -\-  r 

g    '     ' a 

The  deficit  is  caused  by  the  lines  of  force  from  the 
needle  cutting  the  galvanometer  coils  while  the  dis- 
charge is  passing,  and  its  value  is 

Qr. 


The  shunted  ballistic  galvanometer  therefore  gives 
readings  which  are  too  small,  unless  the  duration  of  the 
discharge  is  very  small  compared  with  the  time  of 
vibration  of  the  needle.* 

In  the  case  of  coils  connected  in  parallel  each  hav- 
ing self-inductances,  it  is  difficult  to  assign  a  true  fixed 
value  to  the  self-inductance  of  the  circuit.  In  fact, 
only  under  special  conditions  can  a  single  coil  with 
self-inductance  be  substituted  for  the  coils  in  parallel 
so  as  to  produce  the  same  effect  as  the  latter  upon 
transient  currents  of  every  duration.  These  condi- 

tions   are  fulfilled    when    the    ratio    of   —    is    constant 

R 

for  all  the  coils,and  an  equivalent  coil  may  then  be  sub- 
stituted for  the  parallel  circuits.  In  this  case  the  re- 
sistance of  the  equivalent  coil  must  be 

Rc      Rl     R2      R3 
and  its  self-inductance  must  be 


*  See  Gerard's  Lemons  sur  V  jfrlectricite,  3d  ed.,  Vol.  I.,  p.  213. 


60  ALTERNATING   CURRENTS. 

These  values  make  — -  equal  to  the  constant  value  of 

X-c 

the  ratio  for  the  individual  coils.* 

21.  Rate  of  Work  in  a  Self -Inductive  Circuit  when  the 
Current  is  rising  or  falling.  —  The  effect  of  self-induc- 
tance has  been  compared  with  the  effect  of  inertia  in  a 
moving  solid.  The  inertia  effect  of  water  flowing  in 
a  pipe,  as  suggested  by  Faraday,  also  represents  many 
analogies.  Thus,  on  impressing  an  electric  pressure 
upon  a  circuit,  the  current  does  not  rise  to  its  full  value 
instantly,  but  increases  as  a  logarithmic  function,  the 
constant  of  which  depends  upon  the  self-inductance  of 
the  circuit.  In  the  same  way,  if  pressure  be  exerted 
upon  water  filling  a  pipe,  the  water  cannot  begin  its  full 
flow  instantly  on  account  of  inertia.  If  a  gate  be  sud- 
denly closed  in  the  pipe  after  the  flow  is  fully  under  way, 
the  momentum  of  the  liquid  tends  to  continue  the  flow, 
and  the  gate  suffers  a  severe  blow.  In  the  same  way, 
upon  opening  an  electric  circuit  a  bright  spark  passes 
on  account  of  the  so-called  extra  current  caused  by  the 
tendency  of  self-inductance  to  uphold  the  flow.  It  must 
always  be  remembered  that  the  analogies  between  the 
flow  of  electric  current  and  moving  solids  or  liquids  are 
by  no  means  exact  (Vol.  I.,  p.  1 1),  but  they  are  quite 
useful  in  fixing  the  meaning  of  the  phenomena.  There 
is  a  marked  difference  between  the  effect  of  bends  on 
the  inertia  effect  in  the  pipe  containing  water  and  in  the 
electric  circuit.  Thus,  in  the  electric  circuit,  a  solenoid 
has  much  more  self-inductance  than  has  the  same  wire 

*  Hospitaller's  Trait'e  de  fEnergie  Electrique,  Vol.  I,  p.  496;  Lede- 
boer  &  Maneuvrier,  Academic  des  Sciences,  1887. 


SELF-INDUCTION    AND  CAPACITY.  6  1 

straightened  out.  On  the  other  hand,  bends  in  a  water 
pipe  cause  the  inertia  effect  to  be  absorbed  by  friction. 
Notwithstanding  the  differences,  the  analogies  are 
worthy  of  further  consideration. 

When  water  in  a  pipe  is  set  in  motion  part  of  the 
force    exerted   upon   it    at   any   moment  is  utilized   in 

accelerating  its  mass  f-  —  -),  and  the  remainder  in  over- 
\  at  J 

coming  frictional  resistances  (Av).     That  is, 


where  F  is  the  pressure  exerted,  v  the  instantaneous 
velocity  of  the  water,  M  is  its  mass,  and  A  is  a  con- 
stant. It  is  here  assumed  that  the  frictional  resistance 
is  proportional  to  the  velocity,  which  is  true  only  when 
v  is  small.  When  the  velocity  of  the  water  has  become 
so  great  that  Avl  =  F,  where  v^  is  the  final  velocity,  the 
acceleration  ceases,  and  the  water  continues  to  flow  at  a 
uniform  velocity  v1  as  long  as  the  force  is  applied.  In 
the  case  of  the  electric  circuit,  the  impressed  electric 
pressure  is  expended  in  overcoming  the  counter  electric 


pressure  due  to  self-inductance  f  —  —  J  and  in  causing  the 
current  to  flow  through  the  resistance  (R)  of  the  circuit, 


or  E  =  cR  -f  --     This  is  similar  to  the  expression  for 

dt 
the  flow  of  a  liquid  as  given  above.     cR  represents  the 

electric   pressure   exerted    in    overcoming   the   electric 
resistance   or  electric  friction    of    the   conductor    (the 

,  Ldcf    d(LC}\     . 
active  pressure),  and  —-(=    v         1,  the  pressure  ex- 

erted in  storing  energy  in    the  magnetic   field,   or   in 


62  ALTERNATING   CURRENTS. 

changing  the  momentum  of  the  magnetic  field  (compare 

c  .       _  ..       .      Mdv  (     d(MV}\   .       .       , 

Sect.  19).     Likewise  —-?—  (  =     v  1  in    the   formula 

relating  to  the  flow  of  water  represents,  of  course,  the 
pressure  or  force  exerted  in  storing  energy  in  the  water 
by  increasing  its  momentum.  The  power  expended  in 

the  electric  circuit  at  any  instant  is  EC  =  c2R  H — , 

in  which  c^R  is  the  power  expended  in  heating  the 

conductor  and  Lc —  is  the  power  expended  in  storing 
at 

energy  in  the  magnetic  field.  In  this  discussion,  it  is 
assumed  that  the  electrical  resistance  of  a  conductor  is  a 
constant  and  is  the  same  for  a  variable  current  as  for 
a  constant  one.  This  is  correct  within  practical  limits, 
provided  the  rate  of  variation  of  the  current  is  not  too 
great  and  the  conductor  is  not  too  thick. 

22.    The  Time  Constant  of  a  Self-Inductive  Circuit.  — 
From  the  equation 


dt 

is    given    c  = 73 

dt  R 


which  represents  the  instantaneous  value  of  the  current 
flowing  at  any  moment  while  the  pressure  E  is  applied  to 
a  circuit  of  inductance  L.  To  find  the  value  of  the  in- 
stantaneous current  at  any  particular  time  /,  we  have 

from  the  same  equation,  by  transposition,  — =  — > 

h  —  cK      L 
whence 

Cc__dc__        C'ett 
Jo  E-cR~J*  L' 

i  T      t~Y 

which  gives log  (E  —  cR)     =  —    , 

R  Jo       LJn 


SELF-INDUCTION   AND   CAPACITY.  63 

E-cR\          Rt 


or  log 


E     )       ~  L 


E  -— 

and  finally  c  =  —  (i  —  e  L),  where  e  is  the  base  of  the 
R 

Naperian  logarithms.*  This  shows,  as  already  stated, 
that  the  theoretical  curve  representing  the  rise  or  fall 
of  the  current  is  logarithmic.  The  formula  shows 

that  when  L  is  very  small  .the  current  almost  im- 

p  _K 

mediately  takes  its  full  value  C=—t  since  e  L  quickly 

R 

becomes  negligible  in  comparison  with  unity.  Theoreti- 
cally, when  inductance  is  present,  the  current  can  only 

9L 
rise   to  its  full   value    after   an  infinite   time,  yet  e~^ 

becomes  practically  negligible  after  a  comparatively 
short  interval.  Since  resistance  has  the  absolute  dimen- 
sions of  a  velocity  (a  length  divided  by  a  time)  and 

inductance  has  the  dimensions  of  a  length,  the  ratio  — 

R 
has  the  dimensions  of  a  time  ;   this  ratio,  in  the  case 

of  any  circuit  is,  therefore,  generally  called  the  Time 
Constant  of  the  circuit,  and  may  be  represented  by  the 

Greek  letter  r.     In  the  preceding   equation,  —   repre- 

R 

sents  the  value  which  the  current  would  instantly  reach 
when  under  the  constant  impressed  pressure,  were 
there  no  inductance  in  the  circuit.  This  is  the  same 
as  the  ultimate  value  when  there  is  inductance.  The 
equation  may  therefore  be  written 

_!  —1 

c=C(i—e  T)    or    C  —  c=Ce~*. 

*  See  Gerard's  Lemons  sur  P Electricite,  3d  ed.,  Vol.  I.,  p.  207. 


64  ALTERNATING   CURRENTS. 

When  t  =  T,  this  becomes 


2.718 

This  is  the  deficit  of  the  current  after  a  time  in  sec- 
onds equal  to  r,  and  the  current  at  that  instant  is  there- 
fore .632  of  its  ultimate  or  full  value.  The  value  of 
the  time  constant  is  therefore  a  measure  of  the  growth 
of  the  current  in  a  circuit,  and  it  is  obvious  that  in 
a  circuit  of  great  inductance  and  also  great  resistance, 
the  current  practically  reaches  its  full  value  as  quickly 
as  in  a  circuit  of  small  inductance  and  proportionally 
small  resistance. 

23.  Examples  of  Time  Constants.  —  The  following  are 
the  time  constants  of  some  of  the  circuits  for  which 
inductances  have  previously  been  given  (Sect.  17). 
Wheatstone  bridge  resistances,  when  properly  wound, 
generally  have  a  time  constant  of  a  millionth  of  a  second 
or  less  ;  electric  bell,  4.8  millionths  of  a  second  ;  tele- 
phone call-bell,  nearly  .02  of  a  second  ;  armature  of  a 
small  magneto  generator,  from  .005  to  .013  of  a  sec- 
ond ;  telephone  receiver,  with  diaphragm,  about  .001  of 
a  second  ;  mirror  galvanometer  for  marine  signalling, 
.0016  of  a  second;  mirror  galvanometer  of  5000  ohms 
resistance,  .0004  of  a  second  ;  27ooohm  coil  of  a  mirror 
galvanometer,  .001  of  a  second  ;  ioo,ooo-ohm  coil  of 
a  mirror  galvanometer,  .0007  of  a  second  ;  coil  of 
Ayrton  and  Perry  spring  voltmeter,  .0044  of  a  second  ; 
polarized  relays,  types  I,  2,  3,  and  4,  respectively, 
.0048,  .0045,  .0041,  and  .0052  of  a  second;  Morse 
relays,  from  about  .070  to  .026  of  a  second,  with 


SELF-INDUCTION   AND    CAPACITY.  65 

about  .034  of  a  second  as  an  average  for  instruments 
in  working  adjustment ;  two  telegraph  sounders,  .0095 
and  .0065  of  a  second;  bare  No.  12  B.  and  S.  gauge 
copper  wire  on  a  pole  line,  .00037  °f  a  second  ;  No.  6 
wire  in  a  similar  position,  .0014  of  a  second ;  primary 
of  large  induction  coil,  .09  of  a  second ;  secondary  of 
same,  .065  of  a  second;  dynamo  fields,  from  about  .01 
to  10  seconds  ;  continuous-current  dynamo  armatures, 
from  .005  to  5  seconds ;  Mordey  36-kilowatt  2OOO-volt 
alternator  armature,  .017  of  a  second  ;  Kapp  6o-kilowatt 
2OOO-volt  alternator  armature,  .035  of  a  second  ;  primary 
and  secondary  windings  of  transformers,  from  several 
thousandths  of  a  second  to  a  number  of  seconds. 
Finally,  suppose  6  ohms  is  the  resistance  of  the  mag- 
netizing coil  figuring  in  the  problem  of  Section  16. 
Then  assuming  the  value  of  L  to  be  constant,  which 
is  not  exact  when  the  core  is  iron,  the  time  constant 
becomes  in  the  three  cases,  respectively,  .65,  .92,  and 
.0026  of  a  second. 

24.  Equation  for  Current  in  a  Self-Inductive  Circuit 
when  an  Alternating  Sinusoidal  Pressure  is  applied.  — 
The  total  quantity  of  electricity  which  is  transferred 
through  a  circuit  when  a  periodic  electric  pressure  is 
impressed  upon  it  has  been  shown  to  be  independent  of 
the  inductance  of  the  circuit,  provided  the  period  gives 
sufficient  time  for  the  current  to  follow  its  natural  curve 
of  rise  and  fall ;  the  only  change,  in  this  case,  in  the 
flow  caused  by  inductance  being  a  retardation  of  the 
phase  of  the  current  relative  to  the  pressure  (Sect.  19). 
If,  however,  the  pressure  be  an  alternating  one  the 
quarter  period  of  which  is  not  materially  greater  than 


66  ALTERNATING  CURRENTS. 

the  time  constant  of  the  circuit,  the  current  does  not 
have  an  opportunity  to  gain  its  full  value  before  the 
pressure  falls.  This  causes  a  deficit  in  the  flow,  of  a 
magnitude  depending  upon  the  time  constant  of  the  cir- 
cuit and  the  period  of  the  impressed  pressure  (Sect.  22). 
In  the  case  of  an  alternating  current  set  up  in  a  cir- 
cuit by  an  impressed  alternating  pressure,  this  effect 
reduces  the  current  uniformly  in  each  period.  The 
effect  is  therefore  one  which  makes  an  apparent  in- 
crease in  the  resistance  of  the  circuit. 


Returning  now  to  the  formula  c  = ^-^ Con- 
sidering c  and  E  instantaneous  values  of  current  and 
pressure  which  vary  according  to  a  sine  curve,  and  writ- 
ing for  E  its  value  em  sin  a,  where  em  is  the  maximum 
value  of  the  sinusoidal  pressure;  then 

T(dc\ 
em  sin  a  — LI  —  ) 

c  = — -,  or   dc  H cdt  —  —  sin  adt. 

R  L  L 

It  is  desired  to  find  from  this  equation  the  value  of  c 

=;)  *  * 

this,  the  equation  must  be  integrated.  This  may  be 
most  readily  done  by  assuming  two  arbitrary  variables, 
u  and  v,  the  product  of  which  is  equal  to  c.  Thus 
uv  =  c  and  dc  =  udv  +  vdu.  Substituting  these  values 
for  c  and  dc  in  the  above  formula,  gives 

fvdt       ,  \  fem\ 

u( h  dv\  +  vdu  =  l-^\  sin 


in    terms  of  —    =-       /    e    and  sin  a.     In  order  to  do 

L 


SELF-INDUCTION   AND   CAPACITY.  67 

Since  u  and  v  are  entirely  arbitrary  and  only  their  prod- 
uct is  fixed  by  the  assumed  conditions,  we  are  at  lib- 
erty to  make  further  assumptions  regarding  the  value  of 
one  of  them.  Therefore,  for  further  convenience  in 
integrating,  we  will  assume  such  a  value  for  v  that 

-  —  4-  dv  =  o,  and  the  value  of  v  is  derived  from  this 
T 

by  integration  as  follows:  logv  =  ---  \-logA1,  where 

A'  is  a  constant  of  integration. 

_-  vdt 

Hence  v  —  A'e  T.     Since  -   -  +  dv   is  taken  equal  to 

s? 

zero,  the  principal  equation  reduces  to  vdu  —  —  sin  adt, 

JL/ 

or  dn  =  —  e?  —  sin  aw7/   and   M  =  A"+—,   (  ?—  sin  adt. 
A'      L  A'  J      L 

Whence,  placing  A'  A"  equal  to  A, 

uv  =  c  =e~   A  +J  e?y  sin  adt\ 

-      e     --  C  - 

or,  c  =  Ae  T  +  —e  T  I  €T  sin  a<^. 

Z       «f 

1 
eT  sin  adt  may  be   most  readily  integrated  by  parts  as 

follows  : 

I  eT  sin  adt  =  I  ydz  =^yz  ~  I  zdy* 

t 

Putting    sina=^   and    £*dt=dz,  makes   by  integration 

t 

z  =  re?,  and  by  differentiation  dy  =  cos  ada,  but  a  =  o>/ 
(Vol.  L,  p.  80),  and  therefore  da  =  wdt,  whence 

dy  =  to  cos 


*  See  Price's  Calculus,  Vol.  II.,  p.  358. 


68 


ALTERNATING   CURRENTS, 


Consequently,      J  e^  sin  adt  =  yz  —  J  zdy 


=  re?  sin 


/»    t_ 

—  I  re^o) 


cos 


The  last  term  may  again  be  integrated  by  parts,  putting 

t 
cos  a  =y  and  e*dt  =  dz,  and  the  original  equation   be- 

comes 

/»  «  «  *  /*  i 

j  e^  sin  adt  =  re*  sin  &>/  —  T2o>e?  cos  o>/  —  T2o>2  J  e?  si 

Transposing,  and  substituting  a  for  cot,  gives 

(i  -h  T2o)2)  j  e^  sin  a^  =  rVf-  sin  a  —  w  cos  a  J, 


or         J  6T  sin  a<3f/  = 


6r 


-sin  a  —  a)  cos  a  ).* 

T 


Substituting  the  value  of  this  integral  in  the  expres- 
sion for  the  current,  found  on  the  preceding  page,  gives 


£f4 


/I     .  \ 

I  —  sin  a  —  a)  cos  a  1. 
VT  / 


The  last  term  may  be  put  in  the  form 


sin  a  — 


•  cos  a 


!  See  Price's  Calculus,  Vol.  II.,  p.  84. 


SELF-INDUCTION   AND   CAPACITY. 

.2 
ft) 


69 


Now 


and  this  may  be  written 


I  12 

T 


+ 


12 


ft) 


=  cos2  </>  +  sin2(£, 


where  <£  is  an  angle  whose  cosine  and  sine  equal  re- 
spectively the  first  and  second  terms  in  the  left-hand 
side  of  the  equation.*  Substituting  sin  $  and  cos  <£ 
for  their  equivalents  in  the  last  term  in  the  equa- 
tion for  current  as  developed,  there  results 

(sin  a  cos  <£  —  sin  c£  cos  a) 


sin  (a  -  (f>). 


L+iL 


_-  f> 

Consequently,     c  =  Ae   L  -\ —       m          -sin  (a  —  <£), 


since  —  =  R,  and  co  =  —  =  2  TT/,  where  T  is  the  period 
T  / 

and/=—  is  the  frequency  of   the  alternating  current 

under  consideration.     The  angle  <j>  is  determined  by  the 
condition  that 


, 

tan     = 


sn  (        &> 


*  Chauvenet's  Trigonometry,  p.  90. 


70  ALTERNATING   CURRENTS, 

which  is  obtained  by  dividing 


and   from    this    tan     —  &>r  = 


25.   Exponential  term  is  practically  negligible.  —  The 

exponential  member  of  the  equation  for  the  value  of  c 
shows  the  natural  rise  of  current  when  an  electric 
pressure  is  first  introduced  in  the  circuit.  Its  value 
may  generally  be  entirely  neglected  when  the  press- 
ure is  an  alternating  one,  since  its  effect  becomes 
negligible  within  a  small  interval  after  the  pressure 
is  introduced.  This  is  shown  by  taking  the  current 
equal  to  zero,  as  it  is  at  the  instant  the  pressure  is 
impressed  upon  the  circuit,  and  then  solving  for  the 
value  of  A.  This  is  readily  shown  to  be 

e  ^i 

A  =  --  ,  (=  L  sin  (a,  —  6), 


where  ax  is  the  phase  of  the  alternating  pressure  when 
introduced  in  the  circuit,  and  t^  is  the  time  of  its  intro- 
duction. Substituting  the  value  of  A  in  the  current 


formula  gives 


c  = 


r  R(t-tl)  t  n 

.   sin  (a  —  0)  —  e       L    sin  (04  —  <j>)   • 


-i 

As  t  increases  e  L  quickly  becomes  negligible. 
Therefore,  the  instantaneous  current  due  to  an  alter- 
nating electric  pressure  which  is  impressed  upon  a 
circuit  may  be  ordinarily  taken  to  be 


SELF-INDUCTION   AND   CAPACITY.  71 


c  = 


===  sin  (a  -(/>), 

+  4  7T2/ 2£2 


where  em  is  the  maximum  value  of  the  pressure.     There- 
fore, 7 


where  C  and  .£"  are  the  effective  values  of  current  and 

27T/Z 

pressure.     Since  —  ~  —  =  tan  $, 


R 


^,,        r 
Therefore, 


2  +  4  7T2/2/,2  =  R  Vi  +  tan2  6  =  ~  —r 

™         COS(/) 

emcosd>  „      E 

^   =  --'   _    y  and  6^  = 


A. 


It  is  thus  shown  that  when  a  sinusoidal  electric  press- 
ure is  impressed  in  an  electric  circuit  having  a  constant 
inductance  L,  the  current  is  also  sinusoidal  and  lags  be- 
hind the  pressure  by  an  angle  0,  the  tangent  of  which 

equals  —  j^—  ,  and  which  is  therefore  directly  dependent 

upon  the  inductance  of  the  circuit  and  the  frequency  of 
the  impressed  pressure;  the  maximum  and  effective  cur- 
rents are  less  than  the  maximum  and  effective  pressures 
divided  by  R,  by  an  amount  dependent  upon  the  fre- 
quency and  the  inductance. 

26.  Definition  of  Impedance  and  Reactance.  —  The 
quantity  V^2  +  4  Tr2/2^2  is  generally  called  the  Im- 
pedance of  the  circuit  and  sometimes  its  Apparent  Re- 
sistance, while  2  TT/L  is  sometimes  called  the  Reactance 
or  Inductive  Resistance.  The  square  of  the  impedance 
of  a  circuit  is  therefore  equal  to  the  sum  of  the  squares 
of  its  resistance  and  reactance.  Impedance  and  react- 


72  ALTERNATING   CURRENTS. 

ance  are  both  of  the  dimensions  of  resistance  and  are 
therefore  expressed  in  ohms.  Impedance  may  be  de- 
nned for  self-inductive  circuits  in  general,  as  the  total 
opposition  in  a  circuit  to  the  flow  of  an  alternating  elec- 
tric current,  and  reactance,  as  the  component  of  the 
impedance  caused  by  the  self -inductance  of  the  circuit. 

27.  Circuits  of  Equal  Time  Constants  in  Parallel  and 
Series.  —  The  joint  impedance  of  circuits  combined  in 
parallel  may  be  determined  from  the  impedances  of  the 
individual  circuits,  provided  the  angle  of  lag  is  the 
same  for  all  and  the  circuits  have  no  magnetic  effect 
on  each  other.  Thus,  the  effective  electric  pressure  at 
the  common  terminals  of  the  circuits  is 


E  =  CV^2  +  47T22Z2  =  CR    +  4  7r2/2Z2,  etc. 


Also  E  =  C^R2  +  4  Tr2/2^2,  and  C=C1+CZ  +  etc.  In 
these  expressions  R  and  L  are  the  joint  resistance  and 
apparent  joint  self-inductance,  R-^  Lv  etc.,  are  the  resist- 
ances and  self-inductances  of  the  individual  circuits,  C 
is  the  effective  value  of  the  total  current,  and  Cv  C.2, 
etc.,  are  the  effective  currents  in  the  different  circuits. 
The  above  formulas  may  be  transformed  as  follows  : 


etc 


Adding  these  together  gives 

CI+GI+  etc.  =  C 
E  ~~  E 


H  --  =  =  +  etc. 


SELF-INDUCTION   AND   CAPACITY.  73 


H — ,  +  etc. 

v^  +  ^W 

This  expression  is  similar  to  that  giving  the  joint  resist- 
ance of  divided  circuits,  —  =  —  +  —  +  etc.  The  appar- 

R     R±     R% 
ent  joint  self-inductance  of  this  formula  will  evidently  be 

dependent  upon  the  frequency  except  when  the  time 
constants  of  the  circuits  are  the  same  (compare  Sect. 
20) ;  and  when  the  time  constants  and  therefore  the 
angles  of  lag  of  the  individual  circuits  are  not  equal, 
the  geometrical  sum  instead  of  the  arithmetical  sum  of 
the  reciprocals  of  the  individual  impedances  must  be 
taken  to  get  the  reciprocal  of  the  joint  impedance. 
This  is  fully  developed  later  (Chap.  IV.). 

The  impedance  of  circuits  in  series  is  always  calcu- 
lated from  the  summed  resistances  and  self-inductances, 
provided  the  circuits  have  no  magnetic  effect  on  each 
other  and  contain  no  capacity. 

Thus, 


RI  +  etc.)2  +  4  7T2/2  (^i  +  Aa  +  etc-)2- 

This  is  correct  whether  the  time  constants  of  the  indi- 
vidual circuits  are  equal  or  unequal. 

28.  Triangles  of  Resistance  and  Pressure.  —  In  Sec- 
tion 15  it  is  shown  that  the  impressed  pressure  in  a 
circuit  is  equal  to  the  square  root  of  the  sum  of  the 


74  ALTERNATING   CURRENTS. 

squares  of  the  active  pressure  and  the  pressure  of  self- 
inductance.  Dividing  the  effective  values  of  the  three 
pressures  by  the  current  gives,  in  each  case,  the  equiva- 
lent of  resistance,  so  that  the  three  sides  of  the  triangle 
of  electromotive  forces  are  also  proportional  to  the 
impedance,  reactance,  and  resistance  of  the  circuit  (see 

Fig.  29).  Also,  tan  $  =  — ^ — =-^.  The  electric  press- 
ure of  self-inductance  is  evidently  equal  to  the  rate  at 
which  self-produced  lines  of  force  cut  the  turns  of  a  coil, 
multiplied  by  the  number  of  turns,  or  wL C  =  2  TrfL C. 
Reactance  is  equal  to  the  inductive  pressure  divided  by 
current,  or  2  TrfL  (Fig.  29).  Since  the  line  represent- 
ing active  pressure  lags  behind  that  representing  im- 
pressed pressure  by  the  angle  <£,  the  length  of  the 
former  is  equal  to  that  of  the  latter  multiplied  by  cos  </>, 
or  Ea  =  Et  cos  <f>.  Therefore, 

C  =  ^  =  E>™*,    and   ,.='^t 

exactly  as  shown  by  analysis  (compare  Sect.  25).  The 
triangle  of  electric  pressures  (Fig.  29  «)  and  the  equiva- 
lent triangle  of  resistances  (Fig.  29^),  therefore,  foretell 
the  more  important  of  the  results  that  can  be  gleaned 
from  the  rather  laborious  integrations  which  have  just 
been  performed. 

29.  Application.  —  The  application  to  circuits  in  gen- 
eral, and  to  alternator  armatures  in  particular,  of  the 
deductions  which  are  thus  made  is  evident. 

Thus,  suppose  it  is  desired  to  design  an  alternator 
which  is  to  generate  25  amperes  at  an  effective  press- 
ure of  1000  volts  at  its  terminals,  the  frequency  being 


SELF-INDUCTION   AND   CAPACITY.  75 

100.  Take  first,  for  example,  a  disc  armature  without 
iron  in  its  core,  with  a  resistance  of  I  ohm  and  an 
average  self-inductance  of  .01  henry.  The  effective 
value  of  the  total  pressure  to  be  developed  in  this 
armature  at  full  load  is  then 

^(1000  +  25  x  i)2  4-  (2  TT  x  100  x  .01  x  25)2, 

which  is  equal  to  1037  volts.  Consequently,  the  effect 
of  self-inductance  is  to  demand  an  increase  of  the  total 
pressure  equal  to  12  volts.  Suppose,  however,  the 
armature  is  of  a  type  having  an  iron  core  and  has  an 
average  working  inductance  of  .05  henry,  the  total 
pressure  then  becomes 

\(iooo  -f  25  x  i)2  +  (2  TT  x  100  x  .05  x  25)2, 

which  is  equal  to  1291.  Hence,  the  total  pressure  must 
be  increased  by  266  volts  on  account  of  self-inductance. 
If  the  two  machines  were  worked  at  full  load  upon 
resistances  of  absolutely  no  inductance  or  capacity,  the 
lag  of  the  currents  with  respect  to  the  impressed  press- 
ure in  the  circuit  in  the  two  cases  would  be  respectively 
8°  43',  and  37°  27'  (Fig.  32). 

30.  Resolution  of  an  Irregular  Curve  into  Component 
Sinusoids  or  an  Equivalent  Sinusoid.  —  As  already  said 
(Sect.  5),  it  is  not  safe  to  assume  a  sinusoidal  form  for 
the  curve  of  pressure  developed  by  an  alternator.  In 
general,  it  is  safe  to  say  that  the  curve  produced  by 
nearly  all  machines  having  smooth  core  or  disc  arma- 
tures, is  sufficiently  close  to  a  sinusoid  to  make  the 
deductions  applicable  with  some  degree  of  accuracy. 


ALTERNATING   CURRENTS. 


When  the  curve  does  not  follow  a  sinusoid,  it  is  pos- 
sible to  resolve  it  into  a  number  of  component  sinu- 
soids according  to  Fourier's  theorem,  the  effect  of 
which  can,  to  some  extent,  be  separately  estimated.. 
The  general  analytical  expression  for  the  instantaneous 
current  becomes  c  =  a  sin  a  +  b  sin  2  a  +  c  sin  3  a  •+•  etc. 


LC 


Es=27r/ 


E*=CR 


Fig.  32 

-h  a'  cos  a  +  //  cos  2  a  +  c'  cos  3  a  4-  etc.,  which  is  too 
complex  for  general  use.  The  separation  is  often  more 
readily  effected  by  plotting  sinusoids  by  trial  and  ap- 
proximation (Fig.  33).  The  first  or  fundamental  com- 
ponent sinusoid  has  the  same  period  as  the  primary 
curve,  and  the  other  components  are  regular  harmonics 
pf  the  first.  When  even  harmonics  are  present  the 


SELF-INDUCTION    AND   CAPACITY.  77 

successive  loops  of  the  primary  curve  are  dissimilar, 
but  they  are  similar  when  only  odd  harmonics  are 
present.  Since  the  successive  loops  of  alternating  cur- 
rent curves  are  always  similar,  it  is  evident  that  only  the 
odd  harmonics  need  be  looked  for  in  distorted  curves. 
In  fact,  such  curves  may  nearly  always  be  considered  as 
composed  of  the  fundamental  sinusoid  combined  with 
sinusoids  of  three  times  and  five  times  the  frequency, 
higher  harmonics  being  represented  not  at  all  or  only 
by  a  small  residual.  The  sines  and  cosines  of  the 
Fourier  formula  when  taken  in  combination,  cause  the 


Fig-.  33 

lack  of  symmetry  of  alternating  current  curves.  Even 
if  the  curve  of  pressure  is  an  exact  sinusoid,  if  L  is  not 
absolutely  constant,  the  current  curve  varies  from  the 
sinusoidal  form.  In  circuits  containing  iron  cores,  L 
varies  with  the  value  of  the  current  on  account  of  the 
variations  of  /*,  and  the  current  curve  therefore  takes  an 
irregular  form.  The  amount  of  irregularity  depends 
upon  the  amount  of  iron  present  and  upon  the  extent  of 
its  saturation.  The  expression  for  curves  of  pressure 


78  ALTERNATING   CURRENTS. 

or  current  which  are  not  sinusoidal  may  be  replaced  for 
the  purposes  of  analytical  investigation  by  sinusoidal 
curves  representing  waves  which  give  an  equivalent 
effect.  These  sinusoidal  curves  may  be  called  Equiva- 
lent Sinusoids ;  they  must  have  the  same  frequency  and 
effective  values  as  the  curves  which  they  replace,  and 
their  relative  phase  positions  must  be  such  that  they 
represent  an  equal  amount  of  power.  The  equivalent 
sinusoids  of  curves  that  do  not  vary  much  from  the 
sinusoidal  form  are  practically  the  same  as  the  funda- 
mental harmonic,  but  when  the  primary  curve  varies 
widely  from  the  sine  form  the  equivalent  sinusoid  is 
likely  to  differ  in  both  magnitude  and  position  from  the 
fundamental  harmonic.* 

31.  The  Effect  of  Capacity  in  a  Circuit. —  All  insu- 
lated conductors  have  the  property  of  being  able  to  hold 
electricity  in  its  static  form.  When  such  a  conductor  is 
connected  to  a  source  of  a  different  potential,  electricity 
will  flow  into  or  from  it,  until  its  potential  is  the  same 
as  that  of  the  source.  The  measure  of  the  amount  of 
electricity  which  is  held  by  the  conductor  when  at  unit 
potential  is  its  Capacity,  and  the  C.G.S.  unit  of  capacity 
may  be  defined  as  the  capacity  of  a  conductor  which 
contains  a  unit  charge  of  electricity  when  at  unit  po- 
tential. The  practical  unit  of  capacity  is  the  capacity 
of  a  conductor  which  contains  a  charge  of  one  coulomb 
when  at  a  potential  of  one  volt.  This  is  called  a  Farad, 

*  The  analytical  resolution  of  various  alternator  curves  is  illustrated  by 
Steinmetz  in  Trans.  Amer.  Inst.  E.  E.,  Vol.  12.  The  representation  of 
alternating-current  curves  by  empirical  formulas  is  illustrated  by  Emery  in 
Trans.  Amer.  Inst.  E.  E.,  Vol.  12.  See  also  Appendix  A. 


SELF-INDUCTION   AND   CAPACITY.  79 

after  Faraday,  and  is  — -r  times  as  large  as  the  C.G.S. 

unit  of  capacity.  The  farad  is  too  large  a  unit  of  capac- 
ity to  be  convenient  in  practice,  and  the  microfarad,  or 
millionth  of  a  farad,  is  commonly  used  as  the  unit  of 
measurement.  The  capacity  of  a  conductor  depends 
upon  its  conformation  and  surroundings.  The  term 
Condenser  is  applied  to  any  insulated  conductor  having 
an  appreciable  capacity,  although  it  is  more  strictly  used 
to  designate  a  combination  of  thin  sheets  of  conducting 
material,  insulated,  and  laid  together  with  the  alternate 
layers  connected  in  parallel.  In  the  following  discussion 
the  term  condenser  will  be  used  in  its  broader  sense. 

From  the  foregoing  is  at  once  derived  the  funda- 
mental relation  Q  =  sE,  where  Q  is  the  quantity  of 
electricity  in  coulombs,  s  the  capacity  in  farads,  and 
E  the  potential  in  volts. 

When  a  condenser  is  connected  to  a  source  of  alter- 
nating electric  pressure,  as  indicated  in  Fig.  34,  a  cur- 
rent will  flow  into  and  out  of  the  condenser,  the  value 

CONDENSER 


ALTERNATOR 


Fig-.  34 

of  which  at  any  instant  is  proportional  to  the  rate  of 
change  of  the  active  pressure  (E^ ;  because  the  charge 
in  the  condenser  at  any  instant  is  proportional  to  the 
electrical  pressure  between  the  terminals  of  the  con- 


80  ALTERNATING   CURRENTS. 

denser  at  that  instant,  and  the  rate  at  which  the  charge 
changes  must  be  proportional  to  the  rate  at  which  the 
pressure  changes.  The  rate  of  change  of  the  charge 
is  equal  to  the  number  of  coulombs  flowing  per  second 
into  or  out  of  the  condenser,  and  is  therefore  equal  to 
the  current  flowing  into  or  out  of  the  condenser.  Then 
at  any  instant  the  condenser  current  (ct)  will  be 


- 

•~  dt' 

and  since,  when  the  alternating  pressure  is  sinusoidal, 

—  =  2  irfem  cos  a,   where  em  is  the   maximum   pressure 
dt 

acting  on  the  condenser,  there  results 

cs  =  2  7rfsem  cos  a, 
and  c*     =  em  cos  a  =  es. 

27T/S 

This  pressure,  which  is  in  phase  with  the  condenser 
current,  may  be  called  the  Capacity  Pressure  or  Con- 
denser Pressure.  It  is  90°  in  advance  of  the  active  press- 
ure, as  cos  a  —  sin  (a  +  90°).  That  it  must  be  in  advance 
may  be  readily  seen  from  the  reactions  that  occur  in  the 
circuit.  When  a  sinusoidal  pressure  applied  at  the  ter- 
minals of  a  condenser  is  rising,  a  current  flows  into  the 
condenser.  This  current  is  a  maximum  at  the  instant  the 
pressure  passes  through  zero,  for  at  that  time  the  rate  of 
change  of  pressure  is  a  maximum  (see  Fig.  35  at  point 
N).  When  the  pressure  passes  through  its  maximum 
point,  its  rate  of  change  is  zero,  and  the  current  at  that 
instant  is  zero  ;  when  the  pressure  is  falling,  a  current 
flows  out  of  the  condenser,  If  there  is  appreciable  resist' 


SELF-INDUCTION   AND   CAPACITY. 


8l 


ance  in  the  circuit  the  current  takes  a  short  time  to  build 
up  ;  however,  the  principle  remains  the  same  (Sect.  33). 
From  the  formula  for  instantaneous  current  in  the  con- 
denser the  maximum  current  is  seen  to  be, 


and  the  effective  current 
C.= 


32.  The  Energy  of  a  Charged  Condenser  and  its  Curves 
of  Charge  and  Discharge.  —  As  a  condenser  is  charged, 
a  certain  amount  of  work  is  done  in  raising  the  poten- 
tial of  the  charge.  During  the  time  dt  this  is  equal  to 

dW=Ecdt  = 


or 


=  -qdq, 


(in  which  E  is  a  constant  pressure  impressed  on  the 
condenser  terminals,  c  is  the  current  flowing  into  the 
condenser  at  the  instant  /,  and  q  is  the  final  charge  in 

the  condenser),  from  which,  by  integration,  W=-(-\qi. 

This   represents   a   certain    amount    of  work   which    is 
stored  in  the  condenser  when  its  charge  is  increased 
G 


82  ALTERNATING   CURRENTS. 

from  zero  to  q  coulombs.  When  the  condenser  is  dis- 
charged, an  equal  amount  of  work  is  returned  to  the  cir- 
cuit. The  expression  -I -  }^  is  similar  to  that  giving 

the  work  stored  in,  or  the  kinetic  energy  of,  a  moving 
body  or  an  electro-magnetic  field,  but  the  energy  of  a 
charge  is  truly  potential  and  analogous  to  that  stored  in 
a  compressed  spring.  The  total  work  done  on  a  cir- 
cuit, containing  resistance  and  capacity,  when  a  pressure 
is  impressed  during  a  time  of  charge  dt,  is 

ecdt  =  RPdt  +  -qdq, 

when  e,  c,  and  q  are  the  instantaneous  pressure,  cur- 
rent, and  charge,  and  where  the  last  term  is  the  work 
stored  in  the  charge  dq.  If  this  equation  be  divided  by 
cdt  =  dqt  there  results  an  equation  of  pressure, 


From  these  equations  the  charge  at  any  instant  may  be 
determined  when  the  applied  pressure  e  is  constant 
during  charge ;  and 

~di  +  7 

ri    dq  r*  dt 

Jo  q  —  sE          Jo  Rs 
Integrating, 


log  (?-*£)=  --^4-  log  ,4'; 


hence, 


SELF-INDUCTION   AND   CAPACITY.  83 

solving  for  Af  when  /  =  o,  and  therefore  q  =  o,  there 
results  A'  —  —  sE  =  —  Q  ; 

hence,  q—  Q  (i  —  e~««). 

During  discharge  the  condenser  pressure  is  zero,  and 
therefore 

dq_      _q_m 

"  dt     Rs ' 
from  which,  by  integration, 

~Rs~ 

t 
or  q  =  Ane"x*. 

Solving  for  A"  when  /  =  o,  and  therefore  q—Q  (the 
total  charge),  there  results  A"  =  Q\  hence, 

q  =  Qe~K. 

From  these  equations,  which  are  exactly  similar  to 
those  for  self-induction  (Sect.  21),  it  is  seen  that  the 
curves  of  charge  and  discharge  are  logarithmic  when 
an  unvarying  pressure  is  applied  to  the  system  (see 
Fig.  36  a  and  b).  In  many  cases  of  practice  Rs  is  so 
small  that  the  charge  and  discharge  of  a  condenser  are 
practically  instantaneous. 

33.    Time  Constant  of  a  Circuit  containing  Capacity. — 
In  these  equations  Rs  has  the  same  relation  as  —  in 

the  similar  equations  for  self-inductance  (Sect.  22),  and 
therefore  Rs  may  be  termed  the  time  constant  of  the 


84 


ALTERNATING    CURRENTS. 


condenser  and  may  be  represented  by  r'.     Substituting 
T'  for  Rs  in  the  equation  for  charge  gives 


and  when 


SECONDS 

Pig.  36 

which  shows  that  when  t  =  T'  there  is  a  deficit  in  the 
charge  of  .368  of  its  full  value,  this  full  value  being- 
represented  by  Q  =  Es.  It  is  then  seen  that  a  cir- 
cuit containing  capacity  and  resistance  has  a  time  con- 
stant similar  to  the  time  constant  of  self-inductance, 
and  that  it  is  a  measure  of  the  growth  of  the  charge 
in  the  condenser. 


SELF-INDUCTION   AND    CAPACITY.  85 

34.  Equation  for  the  Current  in  a  Circuit  contain- 
ing Capacity  when  an  Alternating  Sinusoidal  Pressure 
is  applied.  —  In  the  case  when  a  sinusoidal  pressure  is 
impressed  upon  the  circuit  the  equation  of  pressure 

t-xt+ii. 

s 

may  be  differentiated,  giving 


and  as  e  =  em  sin  a,  there  results 

cdt        ,       em 

——  +  &  =  —  Cos  ada. 

Ks  K. 

This  is  a  differential  equation  similar  to  that  for  self- 
induction  and  may  be  integrated  in  the  same  manner. 
The  formula  reduces  to  the  practical  form 

c  =  e™       =  sin  (a  +  <£')  +  Ae~*, 


where  <fi  is  the  angle  by  which  the  current  is  in  ad- 
vance of  the  impressed  pressure.  The  tangent  of  (/>' 
is  found  during  the  development  to  be  equal  to 


27T/S 

(see  treatment  on  self-induction,  Sect.  24). 

_  j^ 

The  exponential  term  Ae  Rs  in  the  general  equation 
represents  the  irregularity  due  to  the  fact  that  the 
current  and  impressed  pressure  must  start  at  the  same 
instant.  That  the  term  must  usually  disappear  in  an 
indefinitely  short  time,  in  practice,  may  be  shown  as  in 
Sect.  25  in  a  similar  instance. 


86  ALTERNATING   CURRENTS. 

E 


Then  C  = 


47T2/2*-2 


is  the   effective  current  in  the  circuit,  -\/  /?2  . 

*  4 

is  the  impedance  or   apparent  resistance,   and     —  —  is 

2  TT/S 
the  reactance  due  to  capacity.     The  first  formula  may 

be  written 


from  which  triangles  of  pressures,  similar  to  Fig.  38, 
and  of  resistances  may  be  constructed  (see  Fig.  37  a 
and  b). 

Since  —  -  --  r-  R  =  tan  d>', 

27T/S 

E  E  cos  <>' 


R  Vi+tan2</>'  R 

It  is  thus  shown  that  when  a  sinusoidal  electric  press- 
ure is  impressed  in  an  electric  circuit,  having  a  capacity 
s,  the  current  is  also  sinusoidal,  and  leads  the  pressure 

by   an    angle   <£',  the  tangent  of  which  is  —  -  -  ,  and 

2*fRs 
which  is  therefore  inversely  dependent  upon  the  capacity 

in  the  circuit  and  the  frequency  of  the  impressed  press- 
ure ;  the  maximum  and  effective  currents  in  the  circuit 
are  less  than  the  maximum  and  effective  pressures  divided 
by  R,  by  the  ratio  of  unity  to  cos  <//,  and  the  deficit  is 
therefore  dependent  upon  the  frequency  and  capacity. 

The  triangles  of  pressure  and  resistance  can  be  used 
to  represent  these  relations,  exactly  as  was  illustrated  in 
the  case  of  self-inductive  circuits. 


SELF-INDUCTION    AND   CAPACITY. 


In  Fig.  35  suppose  the  line  OB  represents  EaJ  the 
active    pressure  in    a   circuit    containing   a   condenser, 

CR 


27T/S 


2  7T/S 


Pig1.  37 

then  OC  laid  off  90°  in  advance  of   OB  and  equal  to 

E8  =  — — —  will  represent  the  capacity  pressure. 
2  irfs 

The  resultant,  or  OD,  will  be  the  effective  impressed 
pressure 


Fig.  38 

It  will  be  seen  from  this  figure  that  Ea  leads  Et  and 
is  shorter.  From  the  relations  shown  in  the  construc- 
tion, the  expression  E{  =  ^/Ea  +  E8  may  be  formed,  and 
a  triangle  of  pressure  laid  off  as  in  Fig.  38,  where  the 
angle  cba  shows  the  current  lead. 


88  ALTERNATING   CURRENTS. 

35.  Effect  of  Capacity  and  Self-Inductance  combined 
in  a  Circuit,  and  the  Equation  for  the  Current  flowing 
when  an  Alternating  Sinusoidal  Pressure  is  applied. — 

In  the  preceding  discussions  it  has  been  shown  that 
the  instantaneous  pressure  of  self-induction  when  a 
sinusoidal  pressure,  E,  is  applied  to  the  circuit  is 

el  =  em  sin  (a  —  90°), 
while  the  instantaneous  condenser  pressure  is 

e.  =  em  sin  (a  +  90°). 

It  is  therefore  seen  that  es  and  el  are  directly  opposed, 
and  their  difference  will  express  the  effect  when  both 
are  in  a  circuit.  Hence, 

r     E 

L,  =  — _______^^_ . 


2  TTfsJ 

and  the  instantaneous  current  is 

c  =  6m  sin  (a  - 


27rfs 
in  which  <f>'f  is  an  angle  whose  tangent  is 


27T/J 


/  /  \  9 

In  this  case  \/  A32  +  (  2  TT/Z  —      l     }    is  tne  impedance 
x  V  2  TT/>/     . 

and  (27T/Z  --  -  —  )    is    the    combined    reactance    of 
V     J          27rfsJ 

self-induction  and  capacity. 

_t_ 
The  term  At'^'  may  be  shown  to  disappear  in  a  very 

short  time,  as  has  been  done  in  the  similar  case  under 


SELF-INDUCTION    AND   CAPACITY.  89 

self-induction.      T"  is   the   positive  difference   between 
T  and  r'. 


Since  2  vfL  --        -*-  R  =  tan  $  ', 

V  2.irfs/ 

the  active  pressure  will  be 


As  the  equations  are  similar  to  those  of  self-induction 
and  capacity,  triangles  of  pressure  and  resistance  may 
be  drawn  (see  Sect.  28). 

When   27T/Z,  is  greater  than  —  ^—  ,  the  angle  <£"  is 

27TJS 

positive,  and  the  current  lags  behind  the  pressure,  but 
when  -  is  greater  than  2  TrfL,  the  angle  <j>fl  is  nega- 

27T/S 

tive,  and  the  current  leads  the  pressure.  Finally,  when 
2irfL  =  —  —  ,  the  angle  <j>"  is  zero,  and  the  circuit  acts 

27T/S 

towards  an  alternating  current  as  though  it  contained 
neither  self-induction  nor  capacity,  but  only  resistance  ; 
that  is,  the  self-induction  and  capacity  exactly  neutralize 
each  other.  In  this  case,  the  relation  between  L  and  s 


35  rt.  Effect,  on  the  Transient  State  in  a  Circuit,  of 
Self-  Inductance  and  Capacity  combined.  —  When  an  in- 
ductive coil  of  inductance  L,  is  included  in  a  circuit  of 
resistance  R,  and  a  condenser  of  capacity  s,  is  shunted 
across  a  portion  of  the  circuit  of  resistance  r,  the  fol- 
lowing conditions  are  set  up  : 

The  condenser  is  charged  with  a  quantity  of  electricity 
Q  =  sCr,  where  C  is  the  steady  value  of  the  current. 
Now  if  the  impressed  pressure  be  suddenly  removed,  the 


90  ALTERNATING   CURRENTS. 

condenser  will  discharge,  and  the  quantity  of  electricity 
which  will  pass  from  the  condenser  through  the  part  of 
the  circuit  beyond  its  terminals  is 


At  the  same  time  the  self-inductance  will  cause  a  quan- 
tity of  electricity  to  be  transferred  through  the  circuit 
in  the  opposite  direction,  which  is  equal  to 

a  -LC 
qi~~R 

Hence  the  total  quantity  of  electricity  transferred 
through  the  circuit  is 


and  the  effect  of  the  condenser  is  to  apparently  reduce 
the  self-inductance  by  an  amount  equal  to  the  capacity 
of  the  condenser  multiplied  by  the  square  of  the  resist- 
ance around  which  it  is  shunted. 

36.  Methods  of  measuring  Self-Inductance.  —  While 
considering  self-inductance  and  capacity,  it  is  advisable 
to  discuss  the  various  available  and  practical  methods 
of  measuring  the  magnitude  of  the  inductance  of  cir- 
cuits. These  methods  are  based  upon  a  comparison  of 
the  unknown  inductance,  either  with  a  known  resistance 
or  resistances;  with  a  known  capacity;  or  with  a  known 
inductance.  The  latter  may  be  the  inductance  of  a 
standard  coil,  and  may  have  been  determined  by  com- 
putation or  careful  comparative  measurement. 

I.  Direct  Comparison  with  Resistance  (Joubert's 
Method).  The  unknown  coil  is  inserted  in  an  alternat- 
ing circuit  in  series  with  a  standard  resistance  of  negli- 


SELF-INDUCTION   AND   CAPACITY.  91 

gible  inductance.  This  may  be  gained  by  using  a 
straight  strip  of  German  silver,  or  thin  strips  bent  back 
on  themselves,  separated  by  thin  silk  or  oiled  paper 
for  insulation.  The  pressures  at  the  terminals  of  the 
standard  and  inductive  resistances  are  measured  by 
an  electrometer  or  by  a  high-resistance  voltmeter  of 
negligible  inductance.  Then,  if  the  impressed  pressure 
in  the  circuit  is  approximately  sinusoidal,  the  following 
relation  holds  : 


E~  CR 

where  E^  Rv  and  E,  R  are  the  respective  pressures  at 
the  terminals  and  the  resistances  of  the  inductive  and 
standard  resistances  ;  C  is  the  current  flowing  through 
them  ;  f  is  the  frequency  of  the  circuit  ;  and  L1  is  the 
inductance  to  be  determined.  Hence, 


and  L  2  ~  or  L     - 

'  '    '  22  ^  ~ 


Rl  is  measured  by  means  of  a  Wheatstone  bridge,  or  by 
some  other  usual  method,  and  f  is  determined  from  the 
speed  and  number  of  poles  of  the  alternator  producing 
the  pressure.  The  measurement  of  pressure  at  the 
terminals  of  the  non-inductive  resistance  is  equivalent  to 
measuring  the  current  which  flows  through  the  non- 

inductive   resistance;    for    C=  —     Substituting  C  for 

E  R 

—  in  the  expression  for  L  gives 
R 


92  ALTERNATING   CURRENTS. 

The  current  may  be  measured  by  an  electrodynamome- 
ter,  instead  of  taking  the  pressure  at  the  terminals  of  a 
standard  known  resistance. 

A  modification  of  this  method  may  be  used  to  deter- 
mine the  working  inductance  of  alternator  armatures. 
Thus,  first  measure  the  pressure  at  the  terminals  of  the 
alternator  when  on  open  circuit  and  normally  excited. 
This  measurement  may  be  made  by  a  high-resistance 
voltmeter  of  negligible  inductance,  such  as  a  Weston 
voltmeter  for  alternating  currents,  a  Cardew  voltmeter 
with  a  considerable  non-inductive  resistance  in  series,  or 
some  type  of  electrostatic  voltmeter.  The  latter  follow 
in  general  the  principle  of  the  Thomson  (Kelvin)  quad- 
rant electrometer,  but  are  so  constructed  as  to  be  port- 
able and  direct  reading.  If  the  armature  current  does 
not  have  too  great  a  demagnetizing  effect  on  the  field, 
the  open  circuit  pressure  may  be  taken  as  the  total 
pressure  which  acts  when  the  armature  is  connected  to 
a  circuit ;  that  is,  it  is  the  impressed  pressure.  Hence, 
connect  the  armature  to  a  load  composed  of  a  known 
resistance  with  negligible  or  known  constant  induct- 
ance, and  measure  the  current  which  flows.  Then 


if  the  curves  of  pressure  and  current  are  approximately 
sinusoidal.  From  this  the  inductance  in  the  circuit  is 
found  to  be  ,  p%  _  /-2£>2\i 

z  =  l~ 


If  the  load  be  an  inductive  resistance,  the  value  of  the 
armature  inductance  is  found  by  subtracting  the  known 


SELF-INDUCTION   AND    CAPACITY.  93 

load  inductance  from  the  circuit  inductance  as  deter- 
mined above.  For,  the  inductive  pressure  of  the  load 
is  27rfL'Cand  that  of  the  armature  is  27rfL"C,  while 
the  total  inductive  pressure  is  2  TrfLC,  which  is  equal  to 
the  sum  of  the  other  two.  Hence, 

2irfLC=27rf(L'  +  L")C, 
and  L"=L-L!. 

If  the  armature  reactions  of  the  machine  thus  tested  be 
considerable,  the  value  of  the  inductance  given  is  too 
great,  but  in  their  effect  upon  regulation,  armature  re- 
actions and  inductance  are  inextricably  mixed,  and 
therefore  cannot  be  entirely  separated. 

This  method  of  measuring  inductance  is  a  conven- 
ient one,  as  the  instruments  used  are  an  electrodyna- 
mometer,  or  other  amperemeter  reading  effective  cur- 
rents, and  a  voltmeter  reading  effective  pressures,  which 
are  portable  and  may  be  used  where  convenience  dic- 
tates. The  result  given  by  this  method  is  the  actual 
working  inductance,  which  is  an  important  feature  when 
the  circuit  contains  iron  and  the  inductance  therefore 
depends  upon  the  volume  of  the  testing  current.  On 
the  other  hand,  the  accuracy  of  the  method  is  not  great. 
Under  the  most  favorable  circumstances  an  accuracy  of 
two  or  three  per  cent  is  attainable.  This  is  sufficiently 
close  for  many  purposes  where  the  method  may  be 
advantageously  used.* 

2.  Comparison  with  Resistance  by  Bridge  (Method  of 
Maxwell  and  Rayleigh).  The  coil  of  unknown  self- 
inductance  L  and  resistance  R  is  placed  in  one  arm  of 

*  Compare  London  Electrician,  Vol.  33,  p.  6. 


94 


ALTERNATING   CURRENTS. 


a  bridge  (Fig.  39).  The  other  resistance  arms  of  the 
bridge  are  non-inductive  and  of  values  A,  B,  and  R' . 
The  bridge  is  first  balanced  in  the  usual  way  to  deter- 
mine the  resistance  R.  With  the  balance  for  constant 
currents  retained,  the  galvanometer  key  is  depressed 
before  the  battery  key.  This  causes  a  throw  of  the 
galvanometer  needle  due  to  the  pressure  of  self-induct- 
ance or  reactance  developed  in  the  coil.  When  C  is  the 
current  in  the  coil,  the  quantity  of  electricity  passed 


Fig.  39 

through  the  bridge  coils  due  to  this  pressure,  is  LC 
divided  by  the  resistance  of  the  bridge  network  (see 
Sect.  19).  The  flow  of  this  electricity  is  through  R 
and  R' ,  in  series  with  the  divided  circuit  made  up  of  A 
plus  B  in  parallel  with  G,  where  G  is  the  resistance  of 
the  galvanometer.  The  resistance  of  the  network  is 

therefore  R  +  R'  +  ^A*^L  and  the  quantity  of  elec- 
G-\-A  -\-B 

tricity  in  coulombs  which  passes  through  the  circuit  is 


SELF-INDUCTION    AND   CAPACITY.  95 

*  • 

LC 

&ii?i^G(A  +  B]  ' 


The  proportion  of  this  which  passes  through  the  galva- 
nometer is 

n       G(A+B)-  (A 

'  ' 


provided  the  galvanometer  needle  does  not  move  appre 
ciably  until  the  impulse  is  past  (Sect.  20).     Hence, 

LC  A  +  B 


.  p. 


G  +  A  +  B 


where  K  is  the  ballistic  constant  of  the  galvanometer. 
Now  the  balance  for  steady  currents  is  disturbed  a 
small  amount  by  the  introduction  of  a  small  resistance 
r  in  the  bridge  arm  with  R.  Suppose  C1  is  the  current 
which  now  flows  through  R  ;  the  effect  of  the  disturbance 
of  the  balance  is  the  same  as  though  a  steady  electric 
pressure  C'r,  opposed  to  the  battery  pressure,  had  been 
introduced  into  the  arm  R.  The  current  flowing  through 
the  galvanometer  on  this  account  is 

_  Cr_  _          A+B 
~  *  -      * 


G  +  A+B 

where  K'  is  the  galvanometer  constant  for  steady  cur- 
rents and  8  the  deflection.  From  these  two  equations 
of  flow,  we  get 

Cr=       CL 

K'S      K  sin  10' 


96  ALTERNATING   CURRENTS. 

L.%*±II 

When  the  galvanometer  is  sensitive  and  r  is  made  very 
small,  the  difference  between  C  and  C1  becomes  small 

and  their  ratio  becomes  sensibly  equal  to  unity,  whence 

rKQ 

L  =  — — ,    provided    the    ballistic   throw  is    sufficiently 
K'S  K  T 

small.     Since  the  ratio  -j^  equals  —  (Vol.  L,  pp.    17 

and  39),  this  may  be  written  L  =- — -• 

C  2     . 

When  —  cannot  be  considered  as  unity  its  value  may 

A.  4-  R 

evidently  be  taken  as  equal  to  — : It  is  evident 

A+R  +  r 

that  a  dead  beat  galvanometer  cannot  be  used  in  this 
work.* 

3.  Comparison  of  Two  Self-Inductances  by  Bridge 
(Maxwell's  Method).  The  two  inductive  resistances 
having  self-inductances  L  and  L1  are  connected  in  two 
arms  of  a  bridge,  together  with  variable  resistances 
which  are  non-inductive  (Fig.  40).  We  will  call  the 
resistances  of  these  arms  R  and  R1 '.  The  other  arms 
of  the  bridge  are  non-inductive  and  of  values  A  and  B. 
First  balancing  the  bridge  in  the  usual  way  for  steady 
currents,  the  proportion  R  :  R1  =  A  :  B  is  given.  Now 
the  galvanometer  key  is  depressed  before  the  battery 
key,  and  if  the  ratio  of  the  impedances  of  the  inductive 
arms  is  not  equal  to  the  ratio  of  A  and  B,  the  gal- 
vanometer needle  will  throw.  R  and  R'  must  then  be 
adjusted  until  a  balance  is  obtained  for  transient  cur- 
rents. This  being  done,  the  balance  for  steady  currents 

*  See  Gerard's  Lecons  sur  tElectricite,  3d  ed.,  Vol.  I.,  p.  322,  and  Gray's 
Absolute  Measurements  in  Electricity  and  Magnetism,  Vol.  II.,  p.  477. 


SELF-INDUCTION   AND   CAPACITY. 


must  again  be  gained  by  adjusting  A  and  B.  This 
will  again  disturb  the  balance  for  transient  currents, 
which  must  be  adjusted  by  changing  R  and  R'.  This 
process  of  trial  and  approximation  is  repeated  until  the 


Fig.  40 

balance  exists  for  both  steady  and  transient  currents, 
when 


=  d,  or 
B 


/2        £2 


But 

Hence, 


A* 


Z'2 


L_  =  A 

L'      B 


Various  modifications  of  the  bridge  arrangement  have 
been    made   in   order   to   facilitate   the   balancing,  but 


98  ALTERNATING   CURRENTS. 

under  the  best  circumstances  the  process  is  a  laborious 
one.* 

3#.  (Ayrton  and  Perry's  Standard  Inductance.)  The 
annoyances  incident  to  making  the  adjustments  re- 
quired in  the  preceding  method  may  be  eliminated  by 
making  the  standard  with  an  adjustable  self-inductance. 
In  this  case,  the  bridge  is  balanced  for  steady  currents 
by  adjusting  A  and  B.  The  balance  for  transient  cur- 
rents is  then  gained  without  altering  any  of  the  resist- 
ances by  adjusting  the  value  of  L' .  This  being  done, 

L      A. 
we  have  as  before,  —  =  —    This  method  has  been  quite 

J ^/  jj 

fully  developed  by  Professors  Ayrton  and  Perry,f  who, 
following  the  methods  of  Professor  Hughes  and  Lord 
Rayleigh,  designed  a  very  satisfactory  inductance  stand- 
ard. This  consists  of  three  coils,  two  fixed  side  by 
side  and  the  third  mounted  so  as  to  rotate  within  the 
others  (Fig.  41).  Calling  the  rotating  coil  A  and  the 
others  B  and  C,  evidently  four  arrangements  can  be 
made;  thus,  A  may  be  connected  with  B  alone,  C  alone, 
B  and  C  in  unison,  or  B  and  C  in  opposition.  Each  of 
these  arrangements  gives  a  maximum  inductance  when 
coil  A  lies  within  the  plane  of  coils  B  and  C  and  the 
field  due  to  its  current  reinforces  that  due  to  the  fixed 
coils.  A  smoothly  graded  variation  of  the  inductance 
may  then  be  gained  by  revolving  coil  A  until  a  mini- 
mum value  for  the  arrangement  is  reached  with  A  at 
1 80°  from  its  preceding  position.  By  means  of  a  prop- 


*  Maxwell's  Electricity  and  Magnetism,  2d  ed.,  Vol.  II.,  p.  367;   Gray's 
Absolute  Measurements,  Vol.  II.,  p.  455. 
t  See  Jour.  Inst.  E.  E.,  Vol.  18,  p.  290. 


SELF-INDUCTION   AND   CAPACITY. 


99 


erly  graduated  circle  the  value  of  the  inductance  may 
be  directly  indicated  for  any  position  of  A  in  either 
arrangement.  If  the  ratio  of  the  resistances  of  the  un- 
known coil  and  the  inductance  standard  does  not  give  a 

A 

value  of  —  which  brings  the  required  value  of  Lf  within 
B 

the  range  of  the  standard,  some  non-inductive  resistance 


Fig.  41 

may  be  included  in  one  of  the  arms  R  or  R! .  Thus, 
suppose  the  unknown  inductance  to  be  nearly  ten  times 
as  great  as  the  highest  value  of  the  standard,  then  ad- 

r> 

justing  the  resistances  of  R  and  R'  so  that  — -  is  greater 

AT  ^ 

than  10  makes  —  =  ~  greater  than  10,  and  the  range  of 

x)  _/-/ 

the  standard  inductance  is  sufficient.     A  somewhat  sim- 


100  ALTERNATING   CURRENTS. 

ilar  standard  may  be  readily  made  by  using  two  sole- 
noids that  telescope  each  other. 

4.  Comparison  with  a  Known  Capacity.  In  the  case 
of  condensers,  the  capacity  may  be  said,  in  general,  to 
be  independent  of  the  charge,  that  is,  the  capacity  is  con- 
stant. The  charge  of  a  condenser  (that  is,  the  quantity 
of  electricity  held  by  it)  is  then  directly  proportional  to 
the  difference  of  potential  between  its  plates ;  and  if  the 
condenser  has  impressed  upon  its  terminals  a  transient 
electric  pressure,  its  rate  of  charging  is  proportional  to 
the  rate  at  which  the  pressure  changes.  That  is,  the 
current  flowing  in  a  condenser  is  proportional  to  the 
rate  of  change  of  the  pressure  impressed  upon  it. 
Therefore,  as  the  pressure  rises  the  current  is  flowing 
in  a  positive  direction,  and  when  the  pressure  reaches 
its  maximum  the  current  ceases.  The  phase  of  the 
charging  current  is  consequently  90°  in  advance  of  the 
phase  of  the  pressure  impressed  at  the  condenser  termi- 
nals. If  the  condenser  is  shunted  around  a  non-induc- 
tive resistance,  the  charging  current  is  90°  in  advance  of 
the  active  pressure  which  causes  current  to  flow  through 
the  resistance  around  which  the  condenser  is  shunted. 
In  this  respect  a  capacity  is  exactly  the  opposite  of 
an  inductance.  If  a  conductor  be  shunted  by  a  ca- 
pacity, the  quantity  of  electricity  transferred  in  charging 
the  condenser  during  a  transient  current  is  evidently 
Q  =  sE  =  sCR,  where  s  is  the  capacity,  R  the  resist- 
ance of  the  conductor,  and  C  the  current  flowing  through 
the  latter.  This  quantity  causes  an  apparent  increase 
in  the  current  passing  through  the  conductor,  and  there- 
fore an  apparent  decrease  in  the  resistance  of  the  con- 


SELF-INDUCTION   AND    CAPACITY.  IOI 

doctor  (see  Sect.  31).  Here  again  the  property^of  capac- 
ity is  opposed  to  that  of  inductance,  which 'when  placet  • 
in  a  circuit  increases  its  apparent  resistance  ;tJQ>aJ1;ta,n- 
sient  current.  It  is  therefore  possible  to  practically 
neutralize  the  effect  of  inductance  by  placing  a  proper 
condenser  in  circuit  with  it.  If  the  inductance  varies 
with  the  current,  the  capacity  required  for  neutraliza- 
tion will  also  depend  upon  the  current.  Therefore, 
when  alternating  currents  are  used,  the  neutralization 
may  be  complete  for  the  integral  of  the  current  taken 
over  a  full  period,  while  the  neutralization  is  by  no 
means  complete  at  any  instant.  The  latter  can  be 
effected  only  by  making  a  condenser  with  a  capacity 
which  varies  with  the  charge  in  the  same  way  as  the 
inductance  varies  with  the  current. 

These  relations  between  the  effects  of  a  capacity  and 
of  an  inductance  lead  to  several  methods  of  measuring 
the  value  of  one  in  terms  of  the  other. ,  The  original 
method  suggested  by  Maxwell*  is  as  follows:  The 
unknown  inductance  is  placed  in  the  arm  R  of  a  bridge ; 
the  known  capacity  is  shunted  around  a  variable  non- 
inductive  resistance  in  arm  B ;  and  the  arms  R1  and  A 
are  variable  non-inductive  resistances  (Fig.  42).  By  a 
process  of  trial  and  approximation  similar  to  that  of  the 
third  method,  a  common  balance  is  obtained  for  both 
steady  and  transient  currents,  when  L  =  R1  As  —  RBs. 

This  is  proved  as  follows  :  When  balance  exists  for 
steady  currents,  RB  =  R'A,  while  balance  for  transient 
currents  also  requires  that  at  every  instant 

*  Electricity  and  Magnetism,  2d  ed.,  Vol.  II.,  p.  387. 


102  ALTERNATING   CURRENTS. 

^A~       CB    c  —j-          v^  -         CR. 

Tkec,  pharging  current  of  the  condenser  when  balance 
exists,  is 

_  sBdcB     sAdcA 

L  jy  6  r>    ; =    • 


dt  dt 


Whence 


dt 


Fig.  42 


Ac 

but  CB  =  —~  and  CA  =  cRt  so  that 
h 


B 


dt 


dt  dt 

Since  R'A  -  RB  =  o,    this    becomes   L  =  R1 'As  = 


SELF-INDUCTION   AND   CAPACITY.  103 

The  correctness  of  this  formula  may  also  be  seen  from 
the  fact  that  balance  for  transient  currents  only  holds 
when  the  quantity  of  electricity  transferred  through  the 
non-inductive  half  of  the  bridge  is  increased  by  the 
condenser  by  an  amount  equal  to  the  deficit  which  is 
caused  by  the  inductance  in  the  other  half  of  the  bridge 

times  £,  or  Q  =  «,  =  J  x  |  x | ;   hence,  L  =  RBs* 

The  formula  L  —  RBs  may  be  written  —  =  Bs,  which 

shows  that  the  time  constants  of  the  branches  of  the 
bridge  which  contain  the  inductance  and  capacity  must 
be  equal  when  the  transient  balance  is  obtained. 

4  a.  (Pirani's  Modification.)  To  avoid  the  annoyances 
incident  to  the  adjustment  of  a  simultaneous  balance 
for  steady  and  transient  currents,  the  following  modifi- 
cation of  the  fourth  method  is  advantageous. 

The  three  branches  A,  B,  R'  of  the  bridge  contain 
non-inductive  resistances  only.  The  fourth  branch  con- 
tains the  inductive  resistance  in  series  with  a  non- 
inductive  resistance  r  (Fig.  43).  The  condenser  is 
shunted  around  the  latter.  The  balance  for  steady  cur- 
rents being  obtained,  the  balance  for  transient  currents 
is  gained  by  changing  the  connections  of  the  condenser 
so  as  to  alter  that  portion  of  r  which  is  shunted  by  the 
condenser.  Then,  if  r1  is  the  value  in  ohms  of  that 
portion  (Fig.  43),  L  =  sr'2.  For,  to  give  a  balance  for 
transient  currents  the  charging  current  of  the  con- 
denser must  be  equal  and  opposite  to  the  effect  of  the 
inductance  in  the  circuit.  Hence,  if  x  represent  the 

*  Hospitaller's  Traite  de  V  £nergie  tiledrique,  Vol.  I.,  p.  469. 


ALTERNATING   CURRENTS. 

resistance  of  the  bridge  network  through  which  a  dis- 
charge occurs, 

and  L  =  sr12.* 


If  the  condenser  is  shunted  around  a  portion  i\  of  bridge 
arm  B,  as  suggested  by  Rimington,  the  formula  is 


46.    Another  modification  of  Maxwell's  method  may 
be  made  so  that  it  becomes  quite  convenient  for  use  in 


Fig.  43 

some  cases.  The  bridge  connection  is  made,  omitting 
the  condenser,  and  the  permanent  balance  is  adjusted 
as  before.  Then  the  throw  of  the  needle  is  taken  when 

*  Gerard's  Lemons  sur  l'£.lectricite,  3d  ed.,  Vol.  I.,  p.  324;   and  Hospi- 
talier's  Traite  de  ?£nergie  Electrique,  Vol.  I.,  p.  470.  (Compare  Sect.  35  a.) 


SELF-INDUCTION    AND   CAPACITY.  105 

the  galvanometer  key  is  depressed  first.  This  throw  is 
caused  by  the  effect  of  the  unknown  inductance.  Now 
a  subdivided  condenser  is  connected  as  a  shunt  to  one 
arm  of  the  bridge,  as  in  Maxwell's  method  (Fig.  42), 
and  the  throw  of  the  needle  is  again  taken.  The  throw 
is  now  due  to  the  combined  effect  of  the  condenser 
and  the  inductance,  and  therefore  must  be  numeri- 
cally smaller  than  before,  unless  the  effect  of  the  con- 
denser is  greater  than  that  of  the  inductance,  when  the 
throw  will  be  negative,  and  may  be  numerically  greater 
than  the  inductance  throw.  Another  division  of  the 
condenser  is  now  plugged  into  the  circuit,  and  the 
throw  is  read  as  before.  The  value  of  the  condenser 
which  will  give  a  zero  throw,  or  a  balance,  may  be  deter- 
mined by  interpolation,  when  L  —  RBs,  as  before.* 

In  all  cases  where  condensers  are  used,  it  is  assumed 
that  their  capacities  are  given  in  farads  and  the  resist- 
ances are  given  in  ohms,  in  which  case  the  inductances 
are  found  in  henrys. 

37.'  Use  of  the  Secohmmeter.  —  In  either  of  those 
methods  of  measuring  inductance  which  depend  upon 
a  bridge  balance  for  transient  currents,  there  is  a  cer- 
tain lack  of  sensitiveness.  In  gaining  a  balance  for 
steady  currents  a  very  small  deflection  of  the  needle 
may  be  multiplied  and  so  made  evident  by  properly 
closing  and  opening  the  galvanometer  key.  For  tran- 
sient currents,  however,  the  direction  of  the  throw 
of  the  needle  differs  upon  closing  and  opening  the 
battery  key.  In  order  that  the  multiplying  effect  may 
be  obtained,  it  is  necessary  to  reverse  the  galvanom- 

*  London  Electrician,  Vol.  33,  p.  5. 


io6 


ALTERNATING   CURRENTS. 


eter  terminals  between  each  closing  and  opening.  The 
closing  and  opening  of  the  battery  circuit  (or  what 
is  equivalent,  the  reversal  of  the  battery)  may  be 
effected  in  synchronism  with  the  reversals  of  the  gal- 
vanometer by  means  of  two  commutators  mounted  upon 
a  rotating  shaft.  This  is  in  effect  the  device  designed 
by  Professors  Ayrton  and  Perry,  and  called  by  them 
a  Secohmmeter.*  It  is  shown  diagrammatically  in 
Fig.  44.  The  connections  of  a  bridge  with  standard 
variable  inductance  and  secohmmeter  are  shown  in 
Fig.  45.  When  the  secohmmeter  is  used  in  comparing 
an  inductance,  either  with  another  inductance  or  with  a 
capacity,  the  velocity  at  which  the  commutators  rotate 

does  not  affect  the  result, 
except  to  vary  the  sensibil- 
ity of  the  test,  provided  that 
time  is  given  between  the 
reversals  for  the  current  to 
rise  to  its  full  value.  This 
is  evident  from  the  fact  that 
the  total  quantity  of  elec- 
tricity moved  under  the  in- 
fluence of  self-inductance 
depends  only  upon  the  integral  taken  over  the  current- 
curve  from  zero  to  C,  and  from  C  to  zero,  and  time 
does  not  enter  as  a  factor  of  this  total  quantity.  When, 
however,  the  secohmmeter  is  used  in  the  second  method, 
where  an  inductance  is  compared  with  a  resistance,  the 
number  of  reversals  enters  directly  as  a  factor  of  the 
result.  The  expression  for  the  inductance  is  then 

*  Jour.  Inst.  E.  £.,  Vol.  18,  p.  284;   Electrical  World,  Vol.  13,  p.  232. 


Fig.  44 


SELF-INDUCTION   AND   CAPACITY. 

z  r     A 


IO/ 


where  0  is  the  deflection  when  the  secohmmeter  is 
rotated  at  V  revolutions  per  second,  and  the  bridge 
is  balanced  for  steady  currents,  while  8  is  the  galvanom- 
eter deflection  for  steady  currents  when  the  balance  is 
disturbed  by  altering  B  to  B  +  r.  k  is  a  constant  de- 
pending upon  the  relative  angular  positions  of  the  two 
commutators,  and  can  be  determined  by  calibration. 


Fig.  45 

When  the  secohmmeter  is  used,  the  galvanometer 
may  always  be  dead  beat,  which  gives  an  additional 
advantage  to  its  use  in  the  methods  where  it  is  re- 
quired to  read  the  galvanometer  deflections  for  tran- 
sient currents. 

38.  The  Effect  of  a  Varying  Permeability  in  an  Alter- 
nating-Current Circuit.  —  In  the  theoretical  discussion 
of  this  chapter,  the  counter  electric  pressure  in  an  elec- 


108  ALTERNATING   CURRENTS. 

trie  circuit  due  to  self-induction  has  been  taken  equal  to 

T  /J  V"1 

—  —  ,  L  being  taken  proportional  to  the  permeability  of 

the  magnetic  circuit.  Thus,  if  e1  and  L1  are  the  counter 
electric  pressure  and  self-inductance  of  an  electric  cir- 
cuit without  an  iron  core,  the  formula  gives 


Now,  if  e2  and  L2  are  the  counter  pressure  and  self- 
inductance  for  current  C  when  an  iron  core  is  within  the 
circuit,  the  formula  becomes 


where  /u-  is  the  permeability  of  the  magnetic  circuit 
(compare  Sect.  16).  The  formula  <?2  =  2  is  really 
incorrect,  since  //,  varies  with  C,  so  that  it  should  be 

dC 


In  using  the  formula  e^  =  -~  —  =  fiL1-j-we  have  omitted 

the  effect  on  the  counter  electric  pressure  of  self-induc- 
tion, which  is  caused  by  the  rate  of  change  of  permea- 
bility with  the  current.  The  magnitude  of  this  is 
represented  by  the  term 

Cdp  L,dC        (Cdy\  T  (dC\ 
dC     dt  \pdC)    *\dtJ 

In  general,  it  may  be  said  that  —  -  is  small  compared 

dC 

with  p,  for  the  values  of  the  induction  in  iron  which 


SELF-INDUCTION    AND   CAPACITY.  109 

y^l       J 

are    ordinarily    used    in    practice,    and    — —    is    quite 


small  compared  with  unity.  Under  some  practical  con- 
ditions it  is  possible  that  —~  may  be  as  great  as  4-  or  i 

aC 

of  the  value  of  /x,  but  this  is  not  common.  The  for- 
mulas as  they  have  been  worked  out,  therefore,  may  be 
accepted  as  indicative  of  the  action  in  such  circuits 
containing  iron  cores  as  are  likely  to  be  met  in  actual 
alternating-current  machinery.  The  definition  of  self- 
inductance  adopted  by  the  Chicago  Electrical  Congress 
takes  into  account  the  variability  of  p. 

39.  The  Power  expended  in  a  Circuit  on  which  a 
Sinusoidal  Alternating  Pressure  is  impressed.  —  If  the 
circuit  be  without  inductance  or  capacity  the  current 
wave  agrees  in  phase  with  the  pressure  which  sets  it  up. 
The  rate  of  expenditure  of  energy  in  the  circuit  at  any 
moment  is  equal  to  the  product  of  the  corresponding 
instantaneous  current  and  pressure.  The  average  rate 
of  expenditure  of  energy,  or  the  average  value  of  the 
power  expended,  in  the  circuit  during  a  complete  period 
is  equal  to  the  average  of  all  the  instantaneous  prod- 
ucts. Or, 

IV  =  —  S    ce,  but  'e  =  em  sin  a  and  c  =  — » 
7  R 

where  T  is  the  time  of  a  complete  period  and  em  is  the 
maximum  instantaneous  pressure  ordinate  ;  hence, 

W=—^ce  =  ?± 


T^ 

but          E  =  — =b  and  C  —  —  — 
V2  R 


1  10  ALTERNATING   CURRENTS. 

Hence,  W=-^=CEt 

C  and  E  being  the  effective  values  of  the  current  and 
pressure. 

If  the  circuit  under  consideration  is  reactive,  the  cur- 
rent is  caused  to  lag  behind  or  lead  the  pressure  by  the 
angle  <£.  The  rate  of  expenditure  of  energy  in  the  cir- 
cuit at  any  instant,  is  evidently  still  equal  to  the  product 
of  the  corresponding  instantaneous  values  of  the  cur- 
rent and  pressure.  The  expression  for  the  average 
power  expended  in  the  circuit  is  therefore,  as  before, 


In  this  case,  however,  e  =  em  sin  a,  and  c  =  —  sin  (a  ^  <£) 
(Sect.  24),   where   /  is   the    impedance    of   the    circuit. 

e  2  Cn 

Hence,    W—  —  I     sin  a  sin  (a  T  <£)  da 
TJ-/./O 


7T/ 


m  .  ,         m 

q:  —  -  1-        sm  a  cos  ada  —  —  --  ^, 
TT/     ^°  2  / 

and  since  em  —  E^Tz,  and  -j  =  cm  =  C^2,  there  follows 

W  =  CE  cos  <£.* 
Assuming  the  current  and   pressure  curves  to  have 

*  Compare  Picou's  Machines  Dynamo  £lectriques,  p.  261  ;  Gerard's 
Lemons  sur  riilectridte,  3d  ed.,  Vol.  I.,  p.  224;  Kapp's  Alternating  Cur- 
rents of  Electricity,  p.  46;  etc. 


SELF-INDUCTION   AND   CAPACITY.  in 

equal  positive  and  negative  loops  (Sect.  80),  the  expres- 
sion thus  derived  for  the  power  expended  in  a  circuit 
during  one-half  period  applies  to  every  half  period,  and 
therefore  to  continuous  operation.  In  the  ordinary  meas- 
urement of  current  and  pressure  the  effective  values  of 
the  quantities  are  determined.  Consequently,  the  prod- 
uct of  amperes  and  volts,  thus  determined,  does  not  rep- 
resent the  power  expended  in  a  reactive  circuit,  but  the 
product  must  be  multiplied  by  the  cosine  of  the  angle  of 
lag.  On  the  other  hand,  a  Wattmeter,  that  is,  an  elec- 
trodynamometer  with  one  coil  of  low  resistance  con- 
nected in  series  with  the  circuit  and  another  coil  of 
high  resistance  connected  in  shunt  with  the  circuit, 
averages  the  instantaneous  products,  and  therefore 
gives  readings  that  are  directly  proportional  to  the 
power  absorbed. 

40.  Method  for  Measuring  the  Angle  of  Lag.  —  We 
have  here  a  ready  method  for  determining  the  angle  of 
lag  of  the  current  flowing  in  a  circuit.  Measure  the 
current  flowing  in  a  circuit  by  an  electrodynamometer ; 
measure  the  pressure  at  its  terminals  by  an  electrostatic 
voltmeter  or  some  type  of  non-inductive  voltmeter  of 
very  high  resistance.  Finally,  measure  the  power  ab- 
sorbed in  the  circuit  by  means  of  a  wattmeter  the  press- 
ure coil  of  which  is  non-inductive  and  of  very  high 
resistance.  The  power  in  watts  determined  by  the 
wattmeter  when  divided  by  the  product  of  volts  and 
amperes  gives  the  cosine  of  the  angle  of  lag.  If  the 
curves  of  current  and  pressure  are  of  irregular  form 
this  measurement  will  give  the  angle  of  lag  between 
the  equivalent  sine  curves  (Sect.  30).  We  will  later 


112 


ALTERNATING   CURRENTS. 


take  up  the  effect  of   inductance  in  the  pressure  coil 
of  the  wattmeter  (Sect.  45). 

41.  Blakesley's  Graphical  Proof. — Blakesley  has  given 
a  neat  proof  of  the  formula  W=  CE  cos  </>.*  Returning 
to  the  graphical  representation  of  alternating  pressures 
or  currents  by  means  of  rotating  lines,  let  AB  and  AC 
(Fig.  46)  represent  respectively  the  maximum  value  of 


E' 


Fig.  46 

the  impressed  electric  pressure  in  a  circuit  and  the 
maximum  value  of  the  resulting  current.  The  angle 
BAG  is  the  angle  of  lag.  If  the  Hnes  rotate  about  the 
point  A,  counter-clockwise,  the  instantaneous  projec- 
tions of  the  lines  AB  and  A  C  upon  the  axis  of  Y  repre- 
sent the  instantaneous  values  of  the  pressure  and 


*  Blakesley's  Alternating  Currents  of  Electricity,  2cl  ed.,  p.  6. 


SELF-INDUCTION   AND   CAPACITY.  113 

current,  when  a  is  measured  from  the  X  axis.  It  is 
therefore  desired  to  determine  the  average  value  of  the 
products  of  these  projections.  Draw  AB'  and  AC 
respectively  perpendicular  and  equal  to  AB  and  AC. 
These  lines  represent  the  positions  of  AB  and  AC  after 
revolving  through  90°.  In  the  figure  the  angle  BAX 
represents  a,  and  CAX  represents  a  —  c/>.  Also  the 
angle  B'  AD1  =  BAX,  and  C'AE'  =  CAX.  It  is  then 
seen  from  the  figure  that 

AE  xAD  =  AC  sin  CAX  x  AB  sin  BAX, 
or  cc  =  cm  sin  (a  —  </>)  x  em  sin  a  ; 

and  in  the  same  way 

AE'  x  AD'  =  AC  cos  C'AE'  x  AB'  cos  B]  AD', 
or  c'e1  =  cm  cos  (a  —  <f>)  x  en  cos  a. 

The  mean  of  these  expressions  is 

cc  ~\~  c  '  c'      c  c 

—  =  -2-2-  [sin  a  sin  (a  —  $)  4-  cos  a  cos  (a  —  $)] 

=  £«£=  cos  [a  -  (a  -  <£)]  =  —  cos  $  =  C  E  cos  (/>. 

This  is  the  expression  for  the  mean  of  the  products 
of  e  and  c  for  two  values  of  a  which  are  90°  apart. 
This  mean  value  is  independent  of  the  positions  of  the 
lines  in  the  figure,  and  is  therefore  the  mean  for  all 
positions.* 

*  The  maximum  power  that  can  be  expended  in  an  inductive  circuit 
when  a  given  pressure  is  applied,  may  be  shown  thus  :   W  =  CE  cos  0, 

r>  rf 

and   since    cos  </>  =  —  ,  where   7  is   impedance,  and    C  =  —  ,  there  results 


W  '•=.  -  —  —  •    This  is  a  maximum  when  R  =  2  -nfL.  Hence,  0  =  45°, 


I 


114  ALTERNATING   CURRENTS. 

Power  loops  or  curves  may  be  plotted  as  in  Figs.  47 
to  50,  the  ordinates  of  which  represent  the  products 
of  the  corresponding  ordinates  of  the  current  and  press- 
ure curves.  Figure  47  shows  the  power  loops  for  a  non- 
inductive  circuit  in  which  the  pressure  and  current  re- 
verse their  directions  at  the  same  time,  and  the  power 


ordinates  are  therefore  always  positive  but  their  numer- 
ical value  varies  in  each  half  period  from  o  to  cmem  and 
back  to  o,  so  that  the  power  absorbed  by  the  circuit 

and  the  power  factor  is  70.7  per  cent.  This  expression  for  maximum 
power  is  of  no  practical  importance,  as  in  the  operation  of  electrical  cir- 
cuits and  machinery  the  highest  possible  operating  efficiency  or  plant 
efficiency  is  usually  desired.  A  high  plant  efficiency  is  incompatible  with 
a  low  power  factor. 


SELF-INDUCTION   AND   CAPACITY.  115 

varies  continually  during  each  half  period.  In  this  case 
0  =  o,  cos  0=i,  and  the  average  power  is  W—CE. 
Figure  48  shows  the  power  loops  for  a  reactive  circuit 
in  which  the  angle  of  lag  is  45°.  This  may  be  taken  to 
equally  represent  the  condition  when  the  current  leads 
or  lags.  It  will  be  seen  in  this  case,  that  during  a  por- 
tion of  each  half  period  the  current  and  pressure  are  in 
opposite  directions,  and  some  of  the  ordinates  of  the 


Fig.  48 


power  loops  are  therefore  negative.  This  must  always 
be  the  case  when  the  current  and  pressure  do  not  coin- 
cide in  phase.  During  the  portion  of  the  half  period  in 
which  the  ordinates  of  the  power  loop  are  positive  the 
circuit  absorbs  power,  but  during  the  portion  in  which 
the  ordinates  are  negative  the  circuit  gives  out  power 
which  was  stored  as  magnetic  field  or  condenser  charge, 
and  returns  it  to  the  source.  The  total  energy  given  to 
the  circuit  during  the  half  period  is  equal  to  the  differ- 


n6 


ALTERNATING   CURRENTS. 


ence  of  that  represented  by  the  positive  and  negative 
loops,  and  the  average  power  absorbed  by  the  circuit  is 
equal  to  this  difference  divided  by  the  length  of  the  half 
period.  When  <£  =  45°,  W=  CE  cos  45°  =  .707  CE.  Fig- 
ure 49  shows  the  power  loops  for  a  circuit  in  which  the 
current  and  pressure  differ  in  phase  by  90°.  In  this  case 
the  negative  loops  are  equal  to  the  positive  ones,  or 
the  circuit  and  source  alternately  give  and  take  equal 


/T\ 


w 


amounts  of  energy,  so  that  taking  each  half  period  as  a 
whole,  no  power  is  absorbed  by  the  circuit.  In  this 
case  cos  <£  =  o,  and  therefore  W=  o. 

42.  Definition  of  Power  Factor.  —  The  product  of  the 
effective  values  of  the  current  and  pressure,  CE,  in  a 
reactive  circuit  is  called  the  Apparent  Energy  or  Ap- 
parent Watts  in  the  circuit.  The  reading  of  a  watt- 
meter applied  to  the  circuit,  which  gives  the  value  of 
CE  cos  <£,  gives  the  True  Energy  or  True  Watts  in  the 
circuit.  The  ratio  of  the  true  watts  to  the  apparent 


SELF-INDUCTION   AND   CAPACITY.  117 

watts  in  a  circuit  is  generally  called  the  Power  Factor, 
as  originally  suggested  by  Fleming.  The  power  loops 
for  a  circuit  are  exactly  symmetrical,  provided  the 
original  current  and  pressure  curves  are  sinusoids 
(Figs.  47  to  49).  When  the  pressure  and  current  are 
in  unison  of  phase  the  average  ordinate  of  the  power 
loops  is  equal  to  one-half  of  the  maximum  ordinate, 
since  the  maximum  ordinate  is  equal  to  cmem,  and  the 

C   € 

average   ordinate   is    equal   to    CE  —  ^-^.     When    the 

original  curves  are  not  in  unison  of  phase,  the  average 
power  ordinate  is  equal  to  one-half  of  the  difference 
between  the  maximum  positive  ordinate  and  the  maxi- 
mum negative  ordinate.*  When  the  current  and  press- 
ure curves  are  not  sinusoids,  the  power  loops  are  not 
symmetrical  and  the  average  power  ordinate  does  not 
necessarily  depend  at  all  upon  the  maximum  ordinate 
(Fig.  50).  It  is  evident  from  the  power  loops  that  the 
torque  on  an  alternator  armature  which  is  delivering 
current  to  a  circuit  varies  from  zero  to  a  maximum 
value  which  is  much  greater  than  the  average.  The 
torque  on  a  continuous-current  machine  is  uniform,  and 
the  armatures  of  alternators  are  therefore  subjected  to 
severer  strains  than  are  continuous-current  armatures. 

43.  Wattless  Current.  — The  preceding  expressions 
show  that  the  energy  expended  in  an  inductive  circuit  is 
equal  to  the  effective  value  of  the  impressed  pressure 
and  a  component  of  the  current  which  is  in  phase  with 
the  pressure,  and  has  a  value  of  Ccoscf).  This  may  be 
called  the  Active  or  Working  Current.  The  remaining 

*  Fleming's  Alternate  Current  Transformer,  Vol.  I.,  p.  124. 


Il8         ALTERNATING  CURRENTS. 

component  of  the  current  does  no  work,  and  therefore 
must  be  in  quadrature  with  the  pressure.  This  gives  it 
a  value  of  C  cos  ((/>  -f  90°)  =  C  sin  (f>.  This  component, 
which  does  no  work  during  a  full  period,  is  often  called 
the  Wattless  or  Idle  Current.  For  illustration,  suppose 
in  Fig.  5 1  that  OS  is  a  pressure  applied  to  a  circuit,  OA 
the  current  and  c/>  the  lag  angle.  Resolving  OA  into  its 


Fig.  50 


components,  O  W  and  Of,  in  phase  with  and  at  right 
angles  to  OS,  the  component  OW  multiplied  by  the 
pressure  will  give  the  power  absorbed  by  the  circuit,  and 
OI  will  be  wattless.*  If  <£  were  90°,  the  total  current 
would  be  in  quadrature  with  the  pressure  and  therefore 

*  Thompson's  Dynamo- Electric  Machinery,  4th  ed.,  p.  636. 


SELF-INDUCTION   AND   CAPACITY.  119 

wattless.  While  a  wattless  current  may  do  a  consider- 
able amount  of  work  in  one  quarter  period,  during  the 
next  quarter  period  the  circuit  returns  an  equal  amount, 
and  the  total  work  for  the  period  is  zero.  (Compare 
power  loops.)  A  lag  of  90°  would  only  be  possible  in  a 
circuit  having  no  electrical  resistance,  since  otherwise 
some  energy  would  necessarily  be  expended  in  heating 
the  conductors.  It  is  possible,  however,  to  make  the 
ratio  of  inductive  resistance,  2  TT/Z,  so  great  in  com- 
parison with  the  true  resistance  R,  that  the  lag  is  very 
nearly  90°.  It  is  also  possible  to  make  the  capacity  of  a 
circuit  so  great  in  comparison  with  its  resistance  that  </> 
is  a  lead  of  nearly  90°.  The  latter  condition  is  one  not 

w 


Pig-.  51 

met  in  practice,  but  the  former  may  quite  easily  be 
brought  about  in  circuits  including  underloaded  trans- 
formers of  poor  design. 

The  value  of  the  power  factor  of  a  circuit  is  evidently 
equal  in  numerical  value  to  cos  $,  for,  power  factor  equals 

True  Watts          CE  cos  cf> 

= —  =  COS  (p. 

Apparent  Watts  CE 

The  total  current  in  a  circuit  multiplied  by  the  power 
factor  is,  therefore,  equal  to  the  active  component  of  the 
current.  A  factor,  which  in  the  same  way  is  propor- 


120 


ALTERNATING   CURRENTS. 


tional  to  the  wattless  current,  is  sometimes  called  the 
Induction  Factor  of  a  circuit.  It  is  evidently  equal  in 
numerical  value  to  sin  </>. 

The  following  table,  which  is  similar  to  one  published 
by  Mr.  Emmet,*  gives  the  power  factor  and  induction 
factor  in  a  circuit  for  any  given  lag. 


Lag 

Power 

Induction 

Lag 

Power 

Induction 

Angle. 

Factor. 

Factor. 

Angle. 

Factor. 

Factor. 

±<P 

cos</> 

sii\4> 

*# 

COS0 

sin  <j> 

Degrees. 

Degrees. 

Degrees. 

Degrees. 

O 

1.  0000 

.OOOO 

90 

23 

•9205 

•3907 

67 

I 

.9998 

.0174 

89 

24 

•9135 

.4067 

66 

2 

•9994 

•0349 

88 

25 

.9063 

.4226 

65 

3 

.9986 

•°523 

87 

26 

.8988 

•4384 

64 

4 

.9976 

.0698 

86 

27 

.8910 

•4540 

63 

5 

.9962 

.0872 

85 

28 

.8829 

•4695 

62 

6 

•9945 

.1045 

84 

29 

.8746 

.4848 

61 

7 

•9925 

.1219 

83 

30 

.8660 

.5000 

60 

8 

•9903 

.1302 

82 

31 

.8572 

•5150 

59 

9 

.9877 

.1564 

81 

32 

.8480 

•5299 

58 

10 

.9848 

•!736 

80 

33 

•8387 

•5446 

57 

ii 

.9816 

.1908 

79 

34 

.8290 

•5592 

56 

12 

.9781 

.2079 

78 

35 

.8191 

•5736 

55 

*3 

•9744 

.2249 

77 

36 

.8090 

.5878 

54 

H 

•97°3 

.2419 

76 

37 

.7986 

.60I8 

53 

'5 

.9659 

.2588 

75 

38 

.7880 

.6156 

52 

16 

.9613 

.2756 

74 

39 

.7771 

.6293 

51 

17 

•9563 

.2924 

73 

40 

.7660 

.6428 

5° 

18 

•95  " 

.3090 

72 

4i 

•7547 

.6561 

49 

19 

•9455 

.3256 

71 

42 

•7431 

.6691 

48 

20 

•9397 

.3420 

70 

43 

•73i3 

.6820 

47 

21 

•9336 

•3584 

69 

44     • 

WS 

.6946 

46 

22 

.9272 

•3746 

68 

45 

.7071 

.7071 

45 

sin  </> 

cos<£ 

±<f> 

sin(/> 

COS0 

±</> 

nduction 

Power 

Lag 

'nduction 

Power 

Lag 

Factor. 

Factor. 

Angle. 

Factor. 

Factor. 

Angle. 

*  W.  L.  R.  Emmet's  Alternating  Current  Wiring  and  Distribution. 


SELF-INDUCTION   AND   CAPACITY.  121 

These  deductions  have  all  been  based  on  the  assump- 
tion that  the  pressure  and  current  curves  are  sinusoids. 
It  has  already  been  shown  (Sect.  30)  that  the  current 
curves  in  working  circuits  are  not  likely  to  be  sinusoids, 
though  the  pressure  curves  may  approximate  closely 
thereto.  It  is  then  impossible  to  determine  the  value  of 
the  angle  of  lag  from  the  curves,  since  it  differs  at  the 
zero  and  maximum  points.  Its  equivalent  value  may 
be  determined,  however,  by  using  pressure,  current,  and 
power  readings,  taken  simultaneously,  as  already  ex- 
plained (Sect.  40),  or  by  determining  the  inductance  of 
the  circuit  when  a  working  current  is  flowing,  when  the 

9  irfT 

angle  of  lag  is  deduced  from  the  expression  tan  $  =  —  -  — 

R 

It  may  also  be  determined  by  methods  which  follow. 

44.  Methods  for  measuring  the  Power  in  an  Alternat- 
ing Current  Circuit.  —  It  can  be  readily  understood  that 
the  power  in  an  alternating  current  circuit  may  be 
measured  most  accurately  and  expeditiously  by  means 
of  a  wattmeter.  However,  the  other  methods  here 
given  serve  a  purpose  in  special  cases  or  when  a  watt- 
meter of  the  proper  range  is  not  at  hand. 

I.  Electrometer  Method.  If  the  two  pairs  of  quad- 
rants of  a  quadrant  electrometer  be  connected  with 
points  of  potential,  respectively,  V^  and  V^  and  the 
needle  be  connected  with  a  point  of  potential  F3,  then 
the  deflection  of  the  needle  is  theoretically 


when  k  is  the  constant  of  the  electrometer. 

If  vlt  i>2,  and  z/3  represent  the  instantaneous  values 


122  ALTERNATING  CURRENTS. 

of  the  potential   at   the  points  when  varying   synchro- 
nously as  a  sine  function,  the  deflection  becomes 

k 


If  it  is  desired  to  measure  the  energy  absorbed  by  an 
inductive  circuit  the  electrometer  may  be  used  in  the 
following  manner. 

The  inductive  resistance  BC  is  connected  in  series 
with  the  non-inductive  resistance  AB  (Fig.  52).  Let  the 
potential  of  the  points  A,  B,  and  C  at  any  instant  be 
represented  respectively  by  v^  vv  and  VB,  when  B  is  the 
junction  between  the  inductive  and  non-inductive  resist- 
ance. Then  if  a  quadrant  electrometer  be  connected 
with  its  quadrants  to  A  and  B,  and  its  needle  and  case 
to  C  (Fig.  520),  the  deflection  is 


If  the  connection  of  the  needle  be  interchanged  so 
that  it  is  connected  to  B  while  the  connections  of  the 
quadrants  remain  unchanged  (Fig.  52^),  this  becomes 


By  subtraction,  this  results  in 

k   CT 
d'  -  d  =     ,       (^  -  v£  (e/2  -  7/3)  dt. 


Dividing  this  by  kR,  where  R  is  the  resistance  of  AB, 

gives 

d'  -d 


i  r Tvl  —  v2/          .  . 
=  rJo  "V^    ^ 


SELF-INDUCTION   AND   CAPACITY. 


123 


Now 


R 


is  equal  to  the  instantaneous  value  of  the 


current  passing  through  the  circuit,  and  v^  —  vz  is  the 


Fig.  52 


instantaneous  value  of  the  difference  of  pressure   be- 
tween the  terminals  of  the  inductive  resistance  BC. 

Consequently,  — ~TB~  =  ~^J    cedt  —  W,    where    W  is 


124  ALTERNATING   CURRENTS. 

the  power  absorbed  by  the  inductive  part  of  the  cir- 
cuit.* 

On  account  of  structural  defects,  the  deflections  of 
electrometer  needles  do  not  always  follow  the  theoreti- 
cal law.  Consequently,  it  is  necessary  to  determine  how 
great  the  deviation  is  before  the  instrument  may  be 
relied  upon.f  Or,  the  instrument  may  be  calibrated  by 
the  use  of  continuous  currents  passing  through  known 
resistances,  which  are  so  adjusted  that  vv  v2,  and  ^3  are 
nearly  the  effective  values  of  the  tests. 

I  a.  Electrostatic  Wattmeter.  A  modification  of  the 
quadrant  electrometer  may  be  made  which  reads  directly 
as  a  wattmeter.  \  In  this  case  the  needle  box  is  divided 
diametrically  into  two  parts  instead  of  into  quadrants. 
The  needle  consists  of  a  disc  divided  diametrically  into 
two  parts  (Fig.  53).  The  parts  of  the  circuit  A  and  B  are 
connected  to  the  two  halves  of  the  needle,  and  B  and  C 
to  the  two  halves  of  the  needle  box. 

Then  the  force  which  causes  the  deflection  of  the 
needle  is  theoretically  proportional  to  the  product 


This  instrument  may  also  be  calibrated,  as  explained 
above,  by  passing  a  known  continuous  current  through 

*  Ayrton,  Jour.  Inst.  E.  E.,  Vol.  17,  p.  163  ;  Gray's  Absolute  Measure- 
ments, Vol.  II.,  p.  698  ;  Swinburne,  Note  on  Electrostatic  Wattmeters, 
London  Electrician,  Vol.  26,  p.  571. 

f  Gray's  Absolute  Measurements  in  Electricity  and  Magnetism,  Vol.  II., 
pp.  662  and  699. 

|  Gerard's  Lemons  sur  VElectricite,  3d  ed.,  Vol.  I.,  p.  6n  ;  Hospitalier's 
Traite  de  VEnergie  Electrique,  Vol.  I.,  pp.  205  and  567. 


SELF-INDUCTION    AND   CAPACITY. 


125 


a  known  resistance.  A  wattmeter  of  this  type  has  been 
designed  by  Swinburne  which  may  be  made  direct  read- 
ing or  may  be  read  by  means  of  a  torsion  head  so  that 
the  needles  will  always  remain  in  a  fixed  position  in 
relation  to  the  needle  boxes.* 


Fig.  53 


2.  Three-Voltmeter  Method.^  As  in  the  previous 
method,  a  non-inductive  resistance  must  be  connected 
in  series  with  the  inductive  circuit  to  be  tested. 
(Fig.  54.)  Voltmeters  are  then  respectively  connected 
between  the  points  A  and  By  B  and  C,  and  A  and  C. 
Letting  ev  e2,  and  e  represent  instantaneous  pressures 
at  the  three  voltmeters,  then  e  =  e^  +  e^,  whence 


But  the   instantaneous    value    of   the   power  in    the 
inductive  circuit   is   w  =  ce2  =  -±  e2.       Substituting    the 


*  Swinburne,  Note  on  Electrostatic  Wattmeters,  London  Electrician, 
Vol.  26,  p.  571  ;  Electrical  World,  Vol.  17,  p.  257,  and  Vol.  19,  p.  44. 

t  Suggested  by  Ayrton  and  Sumpner,  London  Electrician,  Vol.  26,  p. 
736  ;  Electrical  World,  Vol.  1 7,  p.  329. 


126 


ALTERNATING   CURRENTS. 


value  of  ^2  already  found  gives  w  =  -^—  (e2  —  e 
and  the  mean  power  absorbed  during  a  period  is 

W=- 


=  -1-  (E*  -  E?  -  Ef), 

2R^ 

where  E,  Ev  and  E2  are  the  respective  readings  of 
the  voltmeters.  If  R  is  not  known  and  the  value  of 
the  current  C  is  known,  the  formula  may  be  written 

1J7  tx        /  r"9  T~"  9  r"  9\ 


2E, 


Fig.  54 


In  order  that  the  results  of  measurements  may  be  the 
most  accurate  possible,  E^  should  equal  Ev  which  makes 
the  method  inconvenient  for  use  in  ordinary  testing. 
Neither  is  the  method  sufficiently  accurate  to  compen- 
sate for  its  disadvantages.  The  accuracy  of  any  par- 
ticular measurement  made  by  this  method  may  be 
checked  by  inserting  a  known  non-inductive  resistance 
in  place  of  the  inductive  circuit. 


SELF-INDUCTION    AND   CAPACITY.  127 

3.  Three- Ammeter  MetJwd*  Instead  of  putting  a 
non-inductive  resistance  in  series  with  the  inductive 
circuit,  it  may  be  placed  in  parallel  with  it.  In  this  case 
amperemeters  must  replace  the  voltmeters  of  the  pre- 
ceding method  (Fig.  55).  One  amperemeter  measures 
the  whole  current  C,  another  measures  the  current  C^ 
in  the  non-inductive  resistance  R,  and  another  measures 
the  current  C%  in  the  inductive  circuit.  It  is  evidently 
essential  that  the  amperemeter  which  is  in  series  with 
the  non-inductive  resistance  shall  be  of  negligible  in- 
ductance. Supposing  c,  c^  c^  be  the  instantaneous  values 


Fig.  55 

of  the  currents  at  any  moment,  we  have  c=cl-\-c2  and 
c2  —  c-f  —  c£  =  2  c^Cy  while 

w  =  ec  -Rcc  --  -(c^-c^-c^ 

1  2  2  2/' 

I     /T  J3_ 

Whence      W=  —  \    wdt  =  —  (C2  —  C?  —  C<?). 

I          Ir.  O          ^  »' 


This  may  also  be  written, 


In  this  case  the  greatest  accuracy  is  given  when  £\  and 

*  Suggested  by  Fleming,  London  Electrician,  Vol.  27,  p.  9  ;  Electrical 
World,  Vol.  1 7,  p.  423. 


128 


ALTERNATING   CURRENTS. 


C2  are  about  equal,  but,  at  the  best,  the  method  is  not 
very  exact.  Its  accuracy  may  be  checked,  as  in  the 
previous  case,  by  replacing  the  inductive  circuit  by  a 
suitable  non-inductive  resistance. 

4.  Other  Three-Instrument  Methods.  Various  modi- 
fications of  the  last  two  methods  have  been  suggested 
by  Ayrton,  Sumpner,  Blakesley,  and  others.*  One  of 
the  obvious  arrangements  is  to  omit  the  amperemeter 


J-^N/S/N/N/V v 

^-^irYirrTT>-(A2/ 


Pig.  56 

in  series  with  the  non-inductive  resistance  of  the  third 
method,  and  connect  a  voltmeter  across  the  circuit  as 
in  Fig.  56.  In  this  case,  the  power  becomes 


5.  Split  Dynamometer  Methods.  If  separate  alter- 
nating currents  of  the  same  frequency  be  passed  through 
the  two  coils  of  an  electrodynamometer,  its  reading 

This   is    equal    to 

L  c/ 

For 


I       /*2" 

will  be  proportional  to  —  I    c^c^dt. 

JL  c/0 


cos 


*  Alternate  Current  and  Potential  Difference  Analogies  in  the  Methods 
of  Measuring  Power,  Phil.  Mag.,  Vol.  32,  p.  204;  London  Electrician, 
Vol.  27,  p.  199;  Electrical  World,  Vol.  18,  p.  131. 


SELF-INDUCTION    AND   CAPACITY.  129 

cl  =  -\/2  £\  sin  a  and  cz  =  V2  C2  sin  (a  —  c£), 

where  </>  is  the  angle  between  the  two  current  waves. 
Substituting,  gives 

i  f  V2<#  =  ^-CTCl  C2  sin  a  sin  (a  -  <j>)  *#, 
_/  «yo  _/  «yo 

which  is  equal  to 

-  1  —  ?  I    sin  a  sin  (a  —  </>)  </a  =  6\  £T2  cos  <£. 

7T        ^O 

An  electrodynamometer  used  in  this  manner  is  called 
a  Split  Dynamometer.  Now  suppose  we  determine  the 
values  of  C±  and  C2  by  means  of  the  dynamometer  used 
as  an  amperemeter  or  by  other  instruments,  then  the 
value  of  cos  c/>  is  at  once  found.  If  the  measurements 
are  all  made  by  the  same  electrodynamometer,  its  con- 
stant does  not  need  to  be  known.  Suppose  the  read- 
ings in  the  two  circuits  are  Cf  =  £3j  and  C22  =  £82,  and 
the  reading  as  a  split  dynamometer  is  C^C^  cos$  =  £S3, 

then   ms  ft  =       3    ,     This  plan  was  first  suggested  by 


Blakesley.* 

5  a.  Blakesley  planned  various  methods  for  using  a 
split  dynamometer  in  measuring  the  power  absorbed  by 
an  inductive  circuit.  f  In  one  of  the  methods  a  non- 
inductive  resistance  is  connected  in  parallel  with  the 
inductive  circuit  to  be  tested  (Fig.  57),  and  a  split  dyna- 
mometer is  connected  so  that  one  coil  carries  the  total 
current,  and  the  other  carries  the  current  of  the  inductive 
branch.  An  amperemeter  is  also  placed  in  the  induc- 

*  Alternating  Currents  of  Electricity,  2d  ed.,  p.  97. 
t  Phil.  Mag.,  Vol.  31,  p.  346. 

K 


130  ALTERNATING   CURRENTS. 

tive  branch.  Calling  c  the  instantaneous  value  of  the 
total  current,  and  cv  c^  respectively,  the  instantaneous 
currents  in  the  non-inductive  and  inductive  circuits,  the 
following  relations  hold  :  the  reading  of  the  split  dyna- 
mometer is  proportional  to  CC^  cos  (f>,  and  that  of  the 


Fig.  57 


amperemeter  gives  C2  ;  but  c^  =  (c  -  c2)  r2  =  cc2  -  c22, 
and  Rcfa  =  R  (cc^  —  c22).  Rc^  is  equal  to  the  instanta- 
neous pressure  between  the  terminals  of  the  non-induc- 
tive resistance,  and  therefore  Rc^  is  equal  to  the 
instantaneous  value  of  the  power  absorbed  by  the 
inductive  circuit.  Integrating  gives 


=  R  (CC2  cos  (/>)  -RCf  =  kRD  -  RQ, 

where  D  is  the  scale  reading  of  the  split  dynamometer, 
and  k  is  its  constant.  Hence,  the  power  absorbed  by 
the  inductive  circuit  is  equal  to  R  times  the  difference 
between  the  reduced  split  dynamometer  reading  and 
the  square  of  the  current  in  the  inductive  circuit. 

A  similar  result  may  be  gained  by  putting  the  ampere- 
meter in  the  non-inductive  branch,  provided  the  instru- 
ment is  non-inductive. 


SELF-INDUCTION   AND   CAPACITY.  131 

6.  Wattmeter  Methods*  Any  instrument  which  di- 
rectly measures  the  true  energy  in  a  circuit  is  called  a 
wattmeter.  The  commonest  form  of  a  wattmeter  is  an 
electrodynamometer  with  one  of  its  coils  connected 
across  the  terminals  of  the  circuit  under  test  and  the 

other  in  series  therewith.     The  electrodynamometer  in 

1  C"T 

its  ordinary  arrangement  measures  the  value  of  —  I    c^dt. 

2  %J  o 

When  arranged  as  a  wattmeter  it  measures  the  value  of 

i   CT 
—  I    cedt,  which  is  evidently  equal  to   CE  cos  <£,  since 

J_  «x  0 

e  —  A/2  E  sin  a  and  c  =  A/2  C  sin  (a  —  </>),  where  <£  is  the 
angle  of  lag  between  the  pressure  and  current  (Sect. 
39).  The  reading  of  a  wattmeter  of  this  type  is 
therefore  directly  proportional  to  the  power,  while  the 
reading  of  the  same  instrument  when  used  as  an  elec- 
trodynamometer is  proportional  to  the  square  of  the  ef- 
fective current.  In  the  usual  arrangement,  wattmeters 
of  this  class  have  a  series  coil  of  a  few  turns  of  thick 
wire,  which  is  placed  in  series  with  the  circuit  to  be 
measured.  The  pressure  coil  is  composed  of  a  few 
turns  of  fine  wire,  which  is  connected  in  series  with  a 
non-inductive  resistance,  and  is  then  connected  across 
the  terminals  of  the  circuit. 

45.  Corrections  to  Wattmeter  Readings.  —  It  is  essen- 
tial that  the  pressure  coil  of  the  wattmeter  be  of 
entirely  negligible  inductance  and  capacity,  or  that 
these  constants  be  so  mutually  adjusted  that  the  time 
constant  is  practically  zero.  If  this  is  not  the  case,  the 
current  in  the  pressure  coil  is  equal  to  — cos  ^\  instead 

I £1 

*  See  also  Method  i  a. 


132  ALTERNATING   CUR-RENTS. 

of  -— ,  where  E  is  the  pressure  in  the  circuit,  (ftx  the 

^i 
angle  of  lag  in  the  pressure  coil  which  is  dependent  on 

9  TrfT 

the  relation  tan  (ftx  =  — £ — 1,  and  Rl  and  Ll  are  respec- 

Ri 
tively  the  resistance  and  self-inductance  of  the  pressure 

coil.  The  currents  in  the  series  and  pressure  coils  now 
have  a  difference  of  phase  which  is  equal  to  $  —  (j>1  in- 
stead of  (ft,  where  (ft  is  the  angle  of  lag  in  the  main 
circuit.  The  reading  of  an  inductive  wattmeter  is 
therefore  proportional  to  CE  cos  (ftt  cos  ((ft  —  (ft^,  while  a 
correct  reading  is  proportional  to  CE  cos  (ft.  The  read- 
ings of  an  inductive  wattmeter  must  therefore  be  multi- 
plied by  a  factor  equal  to 

cos  <ft cos  (ft 

cos  (fti  cos  (<ft  —  (ftj)      cos  (fti  (cos  <ft  cos  (ftx  +  sin  (ft  sin  (ftx)' 

in  order  that  they  may  give  the  true  power.  This  mul- 
tiplier may  be  called  the  " correcting  factor"  of  an 
inductive  wattmeter.  But 


+  4  •K  + 


and  the  correcting  factor  therefore  reduces  to  * 


22 


I+(27T/)2T1 


RR*  +  4  TT'^LL.R,  ~  I  +  (2 


*  Fleming's  Alternate  Current  Transformer,  Vol.  I.,  p.  139;  Gray's 
Absolute  Measurements  in  Electricity  and  Magnetism,  Vol.  II.,  p.  680; 
Loppe  et  Bouquet's  Courants  Alternatifs  Industrie^,  Vol.  I.,  p.  55. 


SELF-INDUCTION    AND    CAPACITY.  133 

The  formulas  show  that  when  rx  is  negligibly  small  (in 
which  case  fa  is  practically  equal  to  zero),  or  TX  is  equal 
to  r  (in  which  case  (f)1  =  </>),  the  correcting  factor  reduces 
to  unity,  and  the  readings  of  the  wattmeter  are  directly 
proportional  to  power.  When  T^  is  less  than  T,  the  cor- 
recting factor  is  less  than  unity,  and  the  wattmeter  reads 
too  high,  and  when  rx  is  greater  than  T,  the  correcting 
factor  is  greater  than  unity,  and  the  wattmeter  reads  too 
low.  The  indications  of  an  inductive  wattmeter  may, 
therefore,  be  either  correct,  too  high,  or  too  low,  de- 
pending upon  the  algebraical  value  of  the  time  constant 
of  the  circuit  upon  which  measurements  are  being  made. 
As  a  general  rule,  the  time  constant,  T,  of  the  circuit  is 
likely  to  be  positive  and  greater  than  that  of  the  watt- 
meter, so  that  the  readings  of  an  inductive  wattmeter 
are  generally  found  in  practice  to  be  too  high ;  but  in 
ordinary  measurements  it  is  impossible  to  determine 
the  value  of  r,  so  that  the  correcting  factor  of  the  watt- 
meter is  unknown.  The  only  safety  in  wattmeter  meas- 
urements of  power  in  alternating-current  circuits,  there- 
fore, lies  in  the  use  of  a  wattmeter  with  such  a  very 
small  time  constant  in  the  pressure  coil  that  it  may  be 
considered  absolutely  negligible. 

Another  correction  due  to  the  power  used  by  the 
wattmeter  itself  is  also  necessary.  Thus,  if  the  press- 
ure coil  be  connected  to  the  circuit  between  the  cur- 
rent coil  and  the  test  circuit  (Fig.  58),  it  is  evident 
that  the  power  measured  includes  that  absorbed  by  the 
pressure  coil.  If  the  current  coil  be  included  between 
the  point  of  connection  of  the  pressure  coil  and  the 
test  circuit  (Fig.  59),  the  power  measured  includes 


134 


ALTERNATING   CURRENTS. 


that  absorbed  by  the  current  coil.  In  either  case  this 
power  should  be  small  and  usually  may  be  neglected, 
but  when  this  is  not  the  case  it  is  easily  determined 
from  the  resistance  of  the  coil  included,  if  the  press- 
ure or  current  is  known.  In  some  wattmeters  a 
special  correcting  coil  wound  over  the  series  coil  is  in- 
troduced in  series  with  the  pressure  coil  which  corrects 


Fig.  58 


Fig.  59 


for  the  current  in  the  pressure  coil,  the  instrument  being 
connected  as  in  Fig.  58.  (Example:  Weston  watt- 
meter.) 

As  in  the  case  of  the  electrodynamometer,  or  other 
instruments  operated  by  electrodynamic  action,  it  is 
necessary  that  a  wattmeter  of  the  type  here  discussed 
shall  have  no  metal  in  its  frame  in  which  foucault  cur- 
rents may  be  developed  (Sect.  73).  If  this  precaution 
is  not  carefully  looked  after,  the  constant  of  the  instru- 
ment will  vary  with  the  frequency,  and  a  calibration  is 
necessary  for  every  frequency.  For  a  properly  built 


SELF-INDUCTION   AND   CAPACITY.  135 

wattmeter,  which  is  used  at  a  point  near  which  there 
are  no  masses  of  metal,  a  single  calibration  with  contin- 
uous currents  is  sufficient. 

46.  The  Spark  caused  by  breaking  a  Self-Inductive 
Circuit.  —  It  is  to  be  expected  (see  Sect.  19)  that  a 
severe  spark  will  pass  upon  breaking  a  circuit  when  it 
is  carrying  a  continuous  current,  if  it  has  a  great  self- 
inductance,  since  the  self-generated  electric  pressure 
tends  to  uphold  the  falling  current.  This  is  indeed  a 
well-known  effect  observed  upon  breaking  circuits  con- 
taining self-inductance.  It  is  seen  in  exaggerated  form 
in  circuits  containing  such  enormous  self-inductances  as 
those  found  in  dynamo  field  windings.  Again,  break- 
ing the  external  circuit  of  a  continuous-current  series 
dynamo  causes  a  much  more  severe  spark  than  break- 
ing the  external  circuit  of  a  shunt  machine.  In  the 
latter  case  the  extra  current,  or  transfer  of  electricity 
due  to  the  self-induction,  flows  from  the  field  coils 
through  the  armature  instead  of  attempting  to  jump 
across  the  break.  It  may  therefore  be  dangerous  to 
break  the  circuit  of  a  series  dynamo  even  while  the 
normal  working  pressure  is  entirely  harmless,  while  no 
special  danger  is  likely  to  come  from  breaking  the  exter- 
nal circuit  of  a  shunt  dynamo.  On  the  other  hand,  it  is 
possible  to  get  an  exceedingly  severe  shock  by  break- 
ing the  field  circuit  of  a  shunt  dynamo  in  which  the 
working  pressure  may  be  quite  low.  The  high  press- 
ure due  to  self-induction  which  is  generated  in  the 
shunt  field  coils  when  the  circuit  is  broken  is  a  fre- 
quent source  of  injury  to  the  insulation.  The  extra 
current,  having  no  outlet,  makes  one  by  jumping  from 


136  ALTERNATING   CURRENTS. 

the  copper  windings  to  the  frame  of  the  machine,  thus 
causing  a  "ground"  or  "burn  out." 

There  are  many  cases  where  it  is  desirable  to  fre- 
quently break  a  continuous-current  circuit  containing  a 
considerable  self-inductance.  It  is  then  necessary  to 
arrange  some  way  of  diminishing  the  spark  at  the 
break  in  order  to  avoid  burning  up  the  break  switch. 
There  are  four  methods  of  reducing  the  spark: 

a.  The  break  may  be  made  gradually  by  introducing 
resistance  into  the  circuit  before  the  switch  is  opened. 
This  resistance  should  vary  gradually  from  zero  to  in- 
finity.     The    manipulation   of   the  resistance  may  be 
caused  by  the  same  motion  which  opens  the  switch. 

This  device  is  used  quite  largely  to  reduce  the  spark 
caused  by  opening  switches  or  automatic  circuit-break- 
ers in  high-pressure  electric-light  or  electric-railway  cir- 
cuits, and  gives  much  satisfaction.  For  this  purpose 
the  switch  carries  an  auxiliary  contact  of  carbon.  This 
contact  is  of  much  higher  resistance  than  the  firm  cop- 
per contact,  and  the  extra  current  spends  its  energy  in 
flowing  through  it.  Therefore  when  the  carbon  con- 
tact is  broken  but  little  spark  passes,  while  what  does 
pass  causes  comparatively  little  burning  upon  a  portion 
of  the  switch  which  may  be  readily  renewed  (Fig.  60). 

b.  A  coil  of  high  resistance  and  wound  in  such  a  way 
as  to  be  fairly  non-inductive  is  placed  in  parallel  with 
the  inductive  circuit  (Fig.  61).     The  resistance  of  the 
non-inductive  coil  may  be  so  great  as  not  to  materially 
alter  the  steady  current  when  the  circuit  is  closed;  but 
when   the   circuit    is    broken   the   extra  current  flows 
around  the  high-resistance  shunt  rather  than  jump  the 


SELF-INDUCTION   AND   CAPACITY. 


137 


break,  and  thus  the  spark  is  reduced  or  entirely  sup- 
pressed. 

c.    The  switch  may  be  shunted  by  a  fine  wire  which 
acts  as  a  fuse.     When  the  switch  is  opened,  breaking 


Fig.  60 

the  circuit,  the  extra  current  spends  itself  by  flowing 
through  the  fine  wire  shunt,  which  it  burns  off  at  the 
same  time.  This  arrangement  makes  it  necessary  to 
replace  the  fuse  before  the  time  of  each  break.  The 


INDUCTIVE 


NON-INDUCTIVE 
Fig".    61 


arrangement  is  used  to  some  extent  upon  the  fuse 
blocks  (Fig.  62)  intended  for  use  in  high-pressure  elec- 
tric-light mains.  The  main  fuse  of  comparatively  low 
resistance  is  shunted  by  a  fine  high-resistance  fuse. 


138 


ALTERNATING   CURRENTS. 


When  the  main  fuse  blows  out,  the  extra  current,  in- 
stead of  causing  a  vicious  spark,  spends  itself  by  flow- 
ing through  the  shunt  fuse,  at  the  same  time  blowing  it 
out. 


Fig.  62 

d.  A  condenser  may  be  so  arranged  that  it  neutralizes 
the  effect  of  the  self-inductance  at  the  time  of  the  break. 
This  may  be  done  in  two  ways  :  i.  The  condenser  may 
be  connected  in  parallel  with  the  inductive  circuit  (Fig. 
63).  Then  upon  the  break  the  capacity  of  the  condenser 


Fig-.  63 

tends  to  neutralize  the  effect  of  the  inductance  since 
the  charging  current  of  the  condenser  due  to  the  rise 
of  inductive  pressure  is  opposite  in  direction  to  the 
extra  current  due  to  self-inductance,  and  the  spark  is 
therefore  reduced  or  suppressed.  2.  The  condenser 


SELF-INDUCTION   AND   CAPACITY. 


139 


may  be  connected  in  parallel  with  the  switch  (Fig.  64). 
In  this  case  the  extra  current  flows  directly  into  the 
condenser,  and  the  spark  is  reduced  or  suppressed. 
The  effect  of  a  condenser  of  fixed  capacity  in  suppress- 
ing the  spark  at  break  due  to  a  fixed  self-inductance  is 
evidently  the  same  in  the  two  positions.  Upon  closing 
the  circuit,  however,  the  condenser  assists  the  rise  of 
current  in  the  circuit  when  in  the  first  position,  but  has 
no  effect  whatever  when  in  the  second  position,  since 
it  is  short-circuited  when  the  switch  is  closed.  A  con- 


Fig-.  64 

denser  is  ordinarily  connected  across  the  terminals  of  the 
primary  circuit-breaker  of  a  Ruhmkorff  induction  coil. 

There  is  a  marked  difference  between  the  amount  of 
spark  ordinarily  produced  upon  breaking  a  continuous 
current  and  an  equal  alternating  one.  For  instance, 
breaking  a  continuous  current  of  25  amperes  at  1000 
volts  pressure  upon  an  ordinary  hand  switch  without 
an  especially  quick  break  is  likely  to  cause  a  lively  arc, 
while  breaking  an  equal  alternating  current  ordinarily 
causes  little  more  than  an  observable  spark.  Some- 


140         ALTERNATING  CURRENTS. 

times,  however,  a  destructive  arc  is  caused  in  breaking 
an  alternating  current.  This  is  particularly  true  when 
fuses  blow  in  high-pressure  alternating-current  mains 
where  the  metallic  vapor  from  the  fuse  serves  as  a  path 
for  the  arc.  This  difference  in  behavior  on  the  part  of 
alternating-current  circuits  is  due  to  the  fact  that  the 
circuit  may  be  broken  at  different  instants  when  the 
current,  and  the  magnetism  set  up  by  it,  have  widely 
different  values. 

47.  The  Self-Inductance  of  Parallel  Wires. —  The  self- 
inductance  of  two  parallel  wires,  hanging  upon  a  pole 
line  or  otherwise,  frequently  introduces  serious  difficul- 
ties into  the  operation  of  long-distance  telephones  or 
telegraphs.  In  the  ordinary  alternating  systems  for 
lighting  and  the  transmission  of  power,  the  effects  are 
not  so  serious,  though  when  the  transmission  is  over  a 
long  distance  the  self-inductance  of  the  line  cannot  be 
neglected.  An  expression  for  the  self-inductance  of 
two  parallel  wires  may  be  developed  thus  :  suppose  that 
two  parallel  conductors  A  and  Af  form  a  circuit  of  in- 
definitely great  length.  Let  C  be  the  current  flowing 
through  the  conductors,  r  their  radius,  and  d  the  dis- 
tance between  their  axes.  Also  let  //.  and  /*'  be  respec- 
tively the  permeability  of  the  medium  surrounding  the 
wires  and  of  the  wires  themselves.  The  strength  of 
the  magnetic  field  (Ha)  at  a  point  outside  of  the  con- 
ductor A  at  a  distance  a  from  its  centre,  and  due  to 
the  current  in  A  is 


*  The  lines  of  force  due  to  the  current  in  the  conductor  are  circles  with 
their  planes  perpendicular  to  the  conductor,  and  if  the  force  at  any  point 


SELF-INDUCTION   AND   CAPACITY.  141 

The  magnetic  induction  (Ba)  at  the  point  is  therefore 


Now  consider  a  space  cut  out  by  two  planes  perpendicu- 
lar to  the  axes  of  the  conductors  and  one  centimeter 


apart  (see  Fig.  65).     Within  this  space  at  a  distance  a 
from  A  a  number  of  lines  of  force 


will  pass  through  a  radial  width  da.  The  total  number 
of  lines  of  force  that  will  pass  between  the  planes  in 
the  distance  between  the  surface  of  A  and  the  centre  of 

A'  will  be, 

7.T        C*  2  &Cda  ^,       d 

Na  = 

at  a  distance  a  from  the  conductor  is  F,  the  work  done  against  it  in  moving 
a  unit  magnet  pole  around  the  conductor  is  W—  2iraF.  Also  by  Vol.  I., 
p.  12,  W=  4irnC,  but  in  this  case,  n  =  I,  and  therefore  2-jraF—  $irC,  or 


142  ALTERNATING   CURRENTS. 

At  any  point  /  within  A  and  at  a  distance  b  from 
the  centre  of  A,  the  magnetic  effect  will  be  as  though 
the  current  within  a  circle  of  radius  b  were  condensed 
at  the  centre,  since  the  magnetic  effect  at  the  point, 
py  of  the  uniform  layer  of  current  in  the  conductor  be- 
yond b  will  be  zero.  The  strength  of  field  at  p  may 
therefore  be  written, 

rr  _2C  ^TTfi  _2Cb 

/7c  ~T~    X    -  9  ~~  o     ? 

b        trr2        r* 
and  the  magnetic  induction  at/, 

E   =  2fJi'Cb 

r* 
Proceeding  as  before, 


and 


But  this  induction  does  not  link  with  the  whole  current, 
but  only  with  that  within  a  circle  of  radius  b.  The 
product  of  the  current  with  the  number  of  lines  of  force 
enclosed  by  it  is 


x        = 


The  self-inductance  of  A  per  unit  length  is  equal  to 
io~9  times  the  number  of  lines  of  force  linking  the  cur- 
rent when  it  has  a  value  of  unity  (Sect.  16),  and  there- 
fore, 


The  effect  of  the  return,  conductor  A1  is  to  exactly  double 


SELF-INDUCTION    AND   CAPACITY.  143 

the  magnetism  which  is  Jinked  or  enclosed  by  the  cur- 
rent, and  therefore  the  self-inductance  per  centimeter 
length  of  circuit  or.  per  two  centimeters  length  of  wire  is 


When  the  conductors  are  of  copper  suspended  in  the 
air,  IJL  =  p'  =  i,  and 


As  a  rule  the  value  of  2  loge  -,  derived  from  the  dis- 

tances apart  and  diameters  of  wires  in  ordinary  elec- 
tric circuits,  is  quite  large  compared  with  J,  and  the 
impedance  of  such  circuits  consisting  of  two  parallel 
wires,  each  /  centimeters  in  length,  may  therefore  be 
approximately  written, 


The  ratio  of  the  impedance  of   a  circuit  to  the  resist- 
ance of  the  conductors  (  —  )  has  been  called  by  Kennelly 


the  Impedance  Factor,  and  its  value  has  been  calculated 
by  him  for  circuits  having  a  wide  range  in  the  values  of 
d  and  r,  and  for  various  frequencies  from  40  to  i4O.f 
Kennelly  has  also  measured  the  resistance  and  im- 
pedance of  a  certain  circuit,  and  found  that  the  actual 
measurements  with  an  approximately  sinusoidal  current 
fully  agree  with  the  computation.^: 

*  Gerard's  Lemons  sur  FElectricite,  Vol.  I.,  p.  232;    Kennelly,  Trans. 
Amer.  Inst.  E.  E.,  Vol.  10,  p.  213. 

t  Kennelly,  Impedance,  Trans.  Amer.  Inst.  E.  E.,    Vol.  10,  p.  175. 
t  Kennelly,  Trans.  Amer.  Inst.  E.  E.,  Vol.  10,  p.  215. 


144  ALTERNATING   CURRENTS. 

Emmet*  has  calculated  a  table  of  the  resistance, 
reactance,  and  impedance  of  circuits  under  various 
conditions  for  the  frequencies  of  60  and  125,  which 
are  frequencies  now  in  general  commercial  use  in  the 
United  States.  The  data  of  Emmet's  table  are  given 
on  page  145.  The  figures  are  calculated  on  the  as- 
sumption of  a  sinusoidal  current,  but  in  practice  the 
current  is  usually  not  sinusoidal,  and  the  actual  react- 
ance and  impedance  are  therefore  likely  to  be  increased 
from  5  to  25  per  cent,  depending  upon  the  elements 
of  the  circuit  and  the  distortions  of  the  current  curve. 
For  average  conditions,  Emmet  advises  adding  15  per 
cent  to  the  figures  of  the  table  on  this  account. 

48.  The  Distribution  of  Current  in  a  Wire. — The  dis- 
tribution of  the  current  over  the  normal  cross-section  of 
a  conductor  along  which  it  flows,  is  uniform,  provided 
the  current  is  steady,  as  this  is  the  distribution  which 
gives  the  least  loss  of  energy  in  the  conductor.  The 
proof  of  this  theorem  is  as  follows:  The  total  power 
lost  in  the  conductor  is,  according  to  Joule's  law,  C2R, 

where  R  is  equal  to  ~£,  /,  p,  and  A  being  the  length, 
A 

specific  resistance,  and  area  of  the  conductor.  Consid- 
ering that  the  conductor  is  divided  into  elementary  fila- 
ments of  equal  area  and  resistance,  r,  and  the  current 
flowing  in  one  of  these  is  c,  then  the  power  lost  in  it  is 
c*r,  and  the  total  power  lost  in  the  conductor  is  SrV, 
which  must  be  equal  to  C2R,  while  ^c  =  C.  These  con- 
ditions can  be  simultaneously  fulfilled  only  when  the 
currents  in  the  filaments  are  all  equal,  or  the  distribu- 

*  Alternating  Current  Wiring  and  Distribution. 


SELF-INDUCTION   AND    CAPACITY. 


145 


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146  ALTERNATING   CURRENTS. 

tion  is  uniform.  In  the  case  of  alternating  currents, 
self-induction  or  "electro-magnetic  inertia"  comes  in 
to  interfere  with  the  uniform  distribution.  Suppose 
the  wire  be  divided  into  elementary  filaments,  then 
the  formula 


which  exhibits  the  number  of  lines  of  force  set  up  by 
current  C,  shows  that  a  greater  number  of  lines  sur- 
rounds the  central  filament  of  the  wire  than  those 
nearer  the  surface.  In  fact,  the  filaments  composing 
the  outside  of  the  wire  are  surrounded  by  //,'  'C  less  lines 
of  force  than  the  central  filament.  When  an  alternating 
current  flows  through  the  wire,  a  counter  electric  press- 

dN 
ure  is  set  up  in  each  filament  which  is  equal  to  —~, 

where  Nf  is  the  number  of  lines  of  force  set  up  by  the 
current  and  which  surround  the  filament  under  consid- 
eration. Since  Nf  increases  from  the  outside  of  the 
wire  towards  the  central  filament,  the  counter  electric 
pressure  is  greatest  at  the  centre  and  least  at  the  sur- 
face of  the  conductors.  Consequently  there  is  a  ten- 
dency for  the  current  to  forsake  the  centre  of  the 
conductor  and  to  take  a  place  nearer  the  surface.  This 
tendency  is  directly  proportional  to  the  frequency  when 
the  current  is  sinusoidal.  It  is  opposed  by  the  ten- 
dency of  the  current  to  a  distribution  which  will  give 
the  least  loss  of  energy,  and  the  current  therefore  dis- 
tributes itself  in  such  a  way  that  the  current  density 
increases  from  the  centre  to  the  surface  of  the  con- 
ductor. This  makes  an  increase  in  the  actual  resistance 
to  the  flow  of  the  current  and  in  the  loss  of  energy  caused 


SELF-INDUCTION   AND   CAPACITY. 


147 


by  the  current  flowing  through  the  conductor.  The  ratio 
of  the  resistance  of  a  conductor  to  an  alternating  cur- 
rent, Ra,  to  its  true  resistance,  or  the  resistance  which 
it  opposes  to  a  continuous  current,  Rc,  may  be  calcu- 
lated by  the  following  formula:* 

i    ft4/4/4 


a_T  , 
Rc~         12    R? 


1 80 


etc. 


When  the  wire  is  copper,  //.  is  equal  to  unity,  and  the 
formula  becomes 


RC 


12 


i  so 


A  table  showing  the   increase  in  the   resistance  of 
wires  when  carrying  alternating  currents  was  first  cal- 


Diameter. 

Area. 

Increase  over 

Fre- 

MM. 

Inches. 

Sq.  MM. 

Sq.  in. 

Ordinary  Resistance. 

quency. 

IO 

•3937 

78.54 

.122 

less  than  Tiff  % 

15 

•5905 

176.7 

.274 

22% 

2O 

.7874 

314.16 

•487 

8% 

25 

.9842 

490.8 

.760 

17!% 

r     80 

40 

1-575 

1256. 

i-95 

68% 

IOO 

3-937 

7854. 

12.17 

3.8  times 

1000 

39-37 

785400. 

1217 

35  times 

9 
13-4 

•3543 
.5280 

63.62 
I4L3 

.098 
.218 

less  than  T17% 

2j% 

L  ioo 

18 

.7086 

254-4 

•394 

8% 

22.4 

.8826 

394-0 

.611 

17%% 

7-75 

•3013 

47-2 

.071 

less  than  T-J-7  % 

1' 

ii.  61 

•  -457° 

106 

.164 

22% 

15.5 

.6102 

189 

.292 

8% 

-33 

19.36 

.7622 

294 

.456 

'7*% 

*  Gray's  Absolute  Measurements  in  Electricity  and  Magnetism,  Vol.  II. 
p.  329;   Gerard's  Lemons  sur  V Electricity  Vol.  I.,  p.  236. 


I48 


ALTERNATING   CURRENTS. 


culated  by  Mordey*  on  data  presented  by  Lord  Kelvin  f 
(see  table  on  preceding  page).  From  this  table  it  is 
seen  that  Ra  is  practically  the  same  as  Rc  for  the  sizes 
of  wire  and  frequencies  which  are  ordinarily  used,  but 
Emmet  J  has  calculated  a  table  which  may  be  conven- 
iently used  in  any  case  where  Ra  differs  from  Rc.  This 
table  is  given  below. 


Product  of  Circular 

Ra 

Product  of  Circular 

Ra 

Mils  and  Frequency. 

Rc 

Mils  and  Frequency. 

RC 

10,000,000 

.OO 

70,000,000 

I.I3 

2O,OOO,OOO 

.OI 

8o,OOO,OOO 

I.I7 

3O,OOO,OOO 

•03 

9O,OOO,OOO 

1.20 

4O,OOO,OOO 

•05 

IOO,OOO,OOO 

1.25 

5O,OOO,OOO 

.08 

I25,OOO,OOO 

i-34 

6O,OOO,OOO 

.10 

150,000,000 

i-43 

This  table  shows  that  the  frequency  or  the  diameter 
of  the  wire  may  be  so  great  that  no  current  at  all  will 
flow  at  the  centre  of  the  conductor,  while  if  the  fre- 
quency is  very  great,  the  current  will  all  remain  at  the 
exact  outer  surface  or  skin  of  the  wire.  Thomson  § 
shows  that  the  value  of  the  current  at  the  distance  x 
from  the  surface  of  a  conductor  is  equal  to 


when  p  is  the  specific  resistance  of  the  material.    Gray  || 
shows  that  however  great  the  diameter  of  a  wire  may 

*  Jour.  Inst.  E.  E.,  Vol.  18,  p.  603. 

f  Jour.  Inst.  E.  E.,  Vol.  18,  pp.  14  and  35. 

\  Alternating  Current  Wiring  and  Distribution. 

§  J.  J.  Thomson's  Elements  of  Electricity  and  Magnetism,  p.  418. 

II  Absolute  Measurements  in  Electricity  and  Magnetism,  Vol.  II.,  p.  338. 


SELF-INDUCTION    AND   CAPACITY. 


149 


be,  its  resistance  to  an  alternating  current  will  never  be 
less  than  the-  true  resistance  of  a  wire  of  the  diameter 
in  centimeters  given  in  the  following  table. 


Frequency. 

Copper. 

Lead. 

Iron,  ju.  =  300. 

80 

1-43 

4.98 

•195 

120 

1.17 

4.08 

•159 

160 

i.  02 

3-52 

.138 

200 

.91 

3-16 

.123 

The  specific  resistance  of  lead  is  not  far  from  that 
of  ordinary  German  silver ;  it  is  about  twice  that  of 
iron,  and  about  twelve  times  that  of  copper.  The 
remarkably  large  skin  effect  in  the  case  of  the  iron  is 
thus  shown  to  be  due  to  its  large  magnetic  permeability. 
The  permeability  of  300  may  be  somewhat  large  to 
assume  for  an  iron  wire  when  under  the  ordinary  cir- 
cuit conditions.  Kennelly  states  that  iron  telegraph 
or  telephone  wires  show  by  measurement  with  small 
currents  that  /JL  is  about  150.  With  increasing  currents 
it  would,  of  course,  increase  to  about  1000,  and  then 
again  diminish. 

The  value  of  the  self-inductance  of  a  wire  was  deter- 
mined in  the  preceding  section  on  the  assumption  of  a 
uniform  distribution  of  the  current  in  the  conductor. 
Any  disturbance  of  this  distribution  on  account  of 
skin  effect  will  reduce  the  value  of  the  self-inductance 
by  a  small  amount.*  The  correctness  of  these  deduc- 


*  Gray's  Absolute  Measurements  in  Electricity  and  Magnetism,  Vol.  II., 


150  ALTERNATING  CURRENTS. 

tions  in  regard  to  self -inductance  and  skin  effect  in  elec- 
trical conductors  is  proved  by  extensive  experimental 
researches  by  Professor  Hughes*  and  Lord  Rayleigh. 

*  The  Self-induction  of  an  Electric  Current  in  Relation  to  the  Nature 
and  Form  of  the  Conductor,  Jour.  Inst.  E.  E.y  Vol.  1 5,  p.  6. 


METHODS    OF    SOLVING   PROBLEMS.  151 


CHAPTER   IV. 

GRAPHICAL    AND    ANALYTICAL    METHODS    OF    SOLVING 
PROBLEMS    IN    ALTERNATING    CURRENT    CIRCUITS. 

49.  Graphical  Methods.  —  Graphical  methods  lend 
themselves  very  satisfactorily  to  the  solution  of  prob- 
lems relating  to  circuits  upon  which  a  sinusoidal  elec- 
trical pressure  is  impressed.  This  application  of  the 
methods  was  first  brought  to  general  attention  by  T.  H. 
Blakesley.*  The  methods  are  those  of  vector  algebra, 
and  are  entirely  analogous  to  those  which  are  so  largely 
used  in  the  graphical  solutions  relating  to  the  composi- 
tion and  resolution  of  forces  in  graphical  statics  and 
relating  to  the  composition  and  resolution  of  velocities, 
etc.,  in  graphical  dynamics.  To  make  the  treatment  as 
simple  as  possible,  the  use  of  the  methods  herein  will 
be  made  to  conform  as  closely  as  possible  to  their  use 
in  the  treatises  on  graphical  statics.  | 

If  the  line  OA  in  Fig.  66  is  conceived  as  swinging 
at  a  uniform  angular  velocity  »  around  the  point  O,  the 
angle  a  which  it  makes  with  the  horizontal  axis  OX  at 
any  instant  is  a  =  wt,  where  /  is  the  interval  of  time  dur- 
ing which  the  line  describes  the  angle  a.  The  instanta- 

*  Blakesley's  Alternating  Currents  of  Electricity. 

t  See  Dubois'  Graphical  Statics,  Hoskins'  Elements  of  Graphic 
Statics,  etc. 


ALTERNATING   CURRENTS. 


neous  projection  Oa  upon  the  vertical  axis  O  Y,  of  the  line 
OA,  has  a  value  Oa  =  OA  sin  a.  If  OA  is  proportional  to 
the  maximum  value  of  a  sinusoidal  function,  its  instan- 
taneous values  are  proportionally  represented  by  the 
instantaneous  projections  of  OA  ;  and  if  OA  is  propor- 
tional to  the  effective  value  of  a  sinusoidal  function,  the 
instantaneous  values  of  the  function  are  proportionally 
represented  by  the  product  of  A/2  into  the  correspond- 
ing instantaneous  projections.  It  is  therefore  possible 
to  represent  all  the  elements  of  a  sinusoidal  function  : 


Fig-.  66 

(i)  by  a  straight  line  which  revolves  at  a  uniform  rate 
around  one  end ;  and  (2)  by  the  instantaneous  projec- 
tions of  the  line.  It  is  evident  that  the  motion  which 
the  projection  of  the  end  A  of  the  revolving  line  makes 
along  the  axis  OY  is  a  simple  harmonic  motion,  and 
that  all  the  theorems  relating  to  simple  harmonic  motion 
may  be  applied  to  these  solutions.  As  is  ordinarily 
done,  the  rotation  of  the  line  will  always  be  considered 
to  be  left-handed  in  the  following  discussions ;  and 
angles  measured  from  right  to  left  will  be  considered 


METHODS    OF   SOLVING   PROBLEMS.  153 

positive,  while  those  measured  from  left  to   right  will 
be  considered  negative. 

If  two  sinusoidal  electric  pressures  of  the  same  fre- 
quency but  having  a  phase  difference  c/>,  act  in  a  circuit, 
the  corresponding  instantaneous  values  are, 


e 
e'  =  ^l~2E'  (sin  a  +  </>). 

The  total  instantaneous  electric  pressure  acting  in  the 
circuit  is  e  +  e'  .     In  Fig.  67  the  pressure  E  is  repre- 

Y 


Fig.  67 

sented  by  the  line  OA,  and  the  pressure  E'  by  the  line 
OAr.  Oa  and  Oa'  are  the  instantaneous  values  of  the 
pressure  for  the  angular  positions  shown.  The  total 
instantaneous  pressure  in  the  circuit  which  corresponds 
to  the  angular  position  shown,  is  equal  to  Oa  -f  Oa',  or 
Oa'f.  It  is  readily  shown  that  Oa"  is  the  projection  of 
the  diagonal  of  the  parallelogram  constructed  upon  OA 
and  OA' .  This  is  true  for  all  angular  positions,  since 


154 


ALTERNATING    CURRENTS. 
X 


K 


i    /  /  \ 

i    /    I  \ 

i  i  /     i  \ 

/     i  *  i 


METHODS    OF    SOLVING    PROBLEMS.  155 

the  sum  of  the  projections  of  the  lines  OA  and  OA ' 
must  be  equal  to  the  sum  of  the  projections  of  the  lines 
OA  and  A  A",  which  in  turn  is  equal  by  construction 
to  the  projection  of  the  diagonal  OA". 

The  length  of  the  line  OA",  therefore,  proportionally 
represents  the  magnitude  of  the  effective  or  maximum 
total  electrical  pressure  in  the  circuit,  and  its  position 
relative  to  that  of  OA  and  OA' ,  represents  the  relative 
phase  position  of  the  total  pressure.  If  instead  of  two 
pressures  acting  in  a  circuit,  there  are  three  or  more, 
as  OA,  OA',  OA",  OA'",  and  OA"",  in  Fig.  68,  the 
same  construction  is  used.  Thus,  completing  the  par- 
allelogram for  OA  and  OA',  their  resultant  OA  1  is 
found.  Completing  the  parallelogram  for  OAl  and  OA" , 
their  resultant  OA%  is  found,  and  again  with  this  and 
OA"'  the  resultant  OA3  is  obtained  ;  finally,  OA±,  the 
final  resultant,  is  obtained  by  combining  OA3  with  OA"". 
The  figure  shows  that  it  is  unnecessary  to  complete  all 
the  parallelograms.  It  is  only  necessary  to  draw  the 
lines  AAV  A^A^  A%A3,  A3A^  respectively  parallel  and 
equal  in  length  to  the  lines  OAf,  OA",  OA'",  and 
OA"",  and  the  line  drawn  from  O  to  the  end  of 
the  last  line  laid  off  gives  the  phase-position  and  mag- 
nitude of  the  total  pressure  in  the  circuit,  regardless  of 
the  number  of  the  components  from  which  it  is  derived 
(Fig.  69).  The  composition  of  electrical  pressures  is 
therefore  exactly  analogous  to  the  composition  of  veloc- 
ities or  of  forces.  As  in  the  case  of  velocities  or  forces, 
the  resultant  of  any  number  of  electrical  pressures  may 
be  determined  by  this  method. 

The  resultant   of  two   sinusoidal  alternating  electric 


i56 


ALTERNATING   CURRENTS. 


currents  which  flow  in  a  divided  circuit  may  be  graphi- 
cally determined  in  the  same  manner.     In  Fig.  67,  let 


OA    and    OA'    be  the  currents   in    the   two  inductive 
branches    of   a  divided   circuit.     The   two    partial   cur- 


METHODS    OF   SOLVING   PROBLEMS.  157 

rents  differ  from  each  other  in  phase  by  an  angle  </>. 
The  instantaneous  values  of  the  currents  are  repre- 
sented by  the  instantaneous  projections  of  the  lines  as 
they  revolve  around  the  point  O.  At  each  instant,  the 
total  current  in  the  main  circuit  is  equal  to  the  sum  of 
the  instantaneous  partial  currents,  or  to  c  +  c1 '.  Conse- 
quently, the  magnitude  of  the  effective  or  maximum 
value  of  the  current  in  the  main  circuit  is  proportion- 
ally represented  by  the  length  of  the  line  OA",  and  the 
angular  direction  of  OArr  gives  the  angular  relation  of 
the  phase  of  the  total  current  to  the  phases  of  the  par- 
tial currents.  When  the  divided  circuit  contains  more 
than  two  branches,  the  same  method  may  be  extended, 
as  already  explained  for  the  composition  of  electric 
pressures. 

For  convenience  in  using  the  graphical  methods  for 
solving  alternating-current  problems,  it  is  well  to  dis- 
tinguish between  two  different  diagrams.  The  first 
diagram  represents  the  magnitude  and  relative  phase 
positions  of  the  electric  pressures  or  currents.  This 
may  be  called  the  Phase  Diagram.  The  other  diagram 
is  the  polygon  formed  by  laying  off  lines  equal  and 
parallel  to  the  lines  in  the  phase  diagram.  This  may 
be  called  the  Vector  Diagram.  Figures  68  (full  lines 
only)  and  69  are  respectively  phase  and  vector  diagrams 
for  representing  five  electrical  pressures.  The  resultant 
pressure  is  represented  in  magnitude  and  phase  by  the 
line  OA±.  If  the  closing  line  of  the  vector  diagram  is 
inserted  in  the  phase  diagram  by  drawing  from  O  a  line 
in  the  direction  obtained  by  following  round  the  vector 
diagram  against  the  direction  in  which  the  lines  were 


158  ALTERNATING   CURRENTS. 

drawn,  the  line  so  inserted  evidently  represents  the  re- 
sultant of  the  component  pressures  or  currents.  If  the 
line  be  drawn  from  O  in  the  opposite  direction,  it  repre- 
sents a  balancing  pressure  or  current. 

These  simple  propositions,  which  so  evidently  come 
from  the  ordinary  graphical  mechanics  (statics  and  dy- 
namics), give  all  the  foundation  that  is  necessary  for 
the  rapid  and  accurate  solution  of  problems  relating  to 
the  flow  of  current  in  simple  and  compound  circuits  con- 
taining definite  resistances,  inductances,  and  capacities 
in  their  different  parts.  For  solutions  of  complicated 
problems  the  graphical  method  is  often  preferable  to 
the  analytical,  because  the  labor  of  analytical  calcu- 
lations is  rapidly  increased  with  the  complication  of  the 
circuits,  while  the  ease  and  accuracy  of  graphical  calcu- 
lations are  not  affected  thereby.  The  graphical  method 
also  has  the  advantage  of  showing  directly  to  the  eye  the 
relative  phases  of  the  pressures  or  currents  in  different 
parts  of  the  circuit.  The  graphical  solutions  have  the 
same  limitations  in  regard  to  alternating  currents  or 
pressures  which  are  not  sinusoidal  as  have  the  analytical 
methods  ;  and  where  the  wave  form  is  not  sinusoidal, 
only  an  approximation  can  be  arrived  at  by  judiciously 
correcting  the  results  shown  by  the  diagrams  based  on 
sine  functions. 

The  problems  relating  to  alternating-current  circuits 
which  may  be  solved  by  graphical  methods  may  be 
divided  into  three  classes:  (i)  where  the  current  flows 
through  all  parts  of  the  circuit  in  series;  (2)  where  the 
same  electrical  pressure  is  impressed  upon  all  parts  of 
the  circuit  (parallel  circuits);  and  (3)  where  the  first  and 


METHODS    OF    SOLVING   PROBLEMS.  159 

second  classes  are  combined.  Solutions  in  the  third 
class  are  effected  by  combining  partial  solutions  of  the 
first  and  second  classes. 

50.  Series  Circuits.  — First  Class.  Suppose  a  circuit  is 
given  which  has  a  certain  resistance  and  self-inductance 
and  it  is  desired  to  know  what  impressed  pressure  with 
a  frequency  /  is  required  to  pass  through  it  a  certain 
current  C.  In  this  case  the  impressed  pressure  is  made 
up  of  two  components  :  (i)  the  pressure  required  to  pass 
the  current  through  the  resistance  of  the  circuit ;  (2)  the 
pressure  required  to  balance  or  overcome  the  counter 
inductive  pressure.  The  inductive  pressure  is  90°  in 
phase  behind  the  active  pressure,  and  the  phase  dia- 
gram which  shows  the  relative  phases  of  the  pressures 
in  the  circuit  is  therefore  like  that  shown  in  Fig.  70. 
The  active  pressure  OA'  is  equal  to  CR,  and  the  induc- 
tive pressure  OA"  is  equal  to  2nrfLC.  The  inductive  com- 
ponent of  the  impressed  pressure  is  required  to  balance 
2  TrfLC  and  is  therefore  equal  to  and  opposite  to  OA" . 
An  arrowhead  is  therefore  placed  on  OA"  to  show  that 
in  the  vector  diagram  its  direction  must  be  taken  from 
A"  to  O,  instead  of  from  O  outwards,  as  is  done  with  the 
other  lines.  The  vector  diagram  is  therefore  given  by 
drawing  O1Al  equal  and  parallel  to  OAf,  A-^A^  equal  and 
parallel  to  A"O,  and  closing  the  polygon  by  the  line 
<9^42  (Fig.  70).  The  line  O^A^  taken  in  the  direction 
from  Ol  to  A<z  represents  the  magnitude  and  relative 
phase  of  the  impressed  pressure.  When  inserted  in  the 
phase  diagram,  it  is  the  line  OA'".  The  angle  <£,  by 
which  the  current  lags  behind  the  impressed  pressure, 
is  the  angle  A^O^A^ 


i6o 


ALTERNATING   CURRENTS. 


If  a  number  of  inductive  circuits  are  connected  in 
series,  the  inductive  pressure  line  of  the  phase  diagram 
is  equal  to  2  rrrfC(L^  +  L2  +  L3  +  etc.),  and  the  active 
pressure  line  is  equal  to  C(Rl  +  R2  4-  R%  +  etc.),  where 

etc.,  are  the  self-induct- 
ances and  resistances  of 
the  different  parts  of 
the  circuit.  If  the  cir- 
cuit is  non-inductive 
the  phase  diagram  and 
vector  diagram  each  be- 
come a  single  horizon- 
tal line  equal  in  length 
to  CR,  while  if  the  in- 
ductive circuit  contains 
2  no  resistance  the  dia- 
grams each  become  a 
single  vertical  line 
which  is  equal  in  length 
to  2  irfLC. 

Dividing  the  lengths 
the  sides  of  the  vec- 
tor polygon  by  the  value  of  C  gives  the  equivalent  resist- 
ance for  each.  The  vector  polygon  may  therefore  be 
used  directly  for  determining  the  impedance  of  a  circuit 
when  the  resistances  and  reactances  of  its  various  parts 
are  known.  A  vector  polygon  representing  pressures, 
and  one  representing  equivalent  resistances,  may  evi- 
dently be  converted,  one  into  the  other,  by  a  simple 
change  of  scale.  It  is  to  be  remembered  that  in  a  series 


METHODS    OF    SOLVING   PROBLEMS.  l6l 

circuit  the  current  has  the  same  phase,  and  is  equal  at 
any  given  instant,  in  all  parts  of  the  circuit,  but  the 
phase,  with  reference  to  the  current,  of  the  pressure  im- 
pressed at  the  terminals  of  different  parts  of  the  circuit 
depends  wholly  upon  the  relation  between  the  individual 
resistances  and  reactances  of  the  parts. 

EXAMPLES.  —  In  the  following  examples  it  is  desired 
to  find  for  each  of  the  given  circuits:  (i)  the  impedance 
of  the  circuit ;  (2)  the  current  which  flows  through  the 
circuit  when  the  impressed  pressure  is  100  volts ; 
(3)  the  impressed  pressure  which  is  required  to  pass 
10  amperes  through  the  circuit;  (4)  the  angle  by  which 
the  current  lags  behind  the  impressed  pressure.  The 
frequency  in  each  case  is  taken  to  be  127^,  whence  2  irf 
is  equal  to  800. 

Circuits  containing  Resistance  and  Self -inductance. 

a.    The  circuit  consists  of  a  non-inductive  resistance 
of    10  ohms.     The  phase  and  vector  diagrams  for  the 
first  two  solutions  are  resistance 
diagrams,    each    consisting   of    a 
horizontal  line  10  units  in  length. 
The  impedance  of  the  circuit  is 
10  ohms,  and  the  current  which 
flows    when    a    pressure    of    100    ? .    ~  inft 1 

OT  O  ri  ^  J.vU 

volts  is  impressed  on  the  circuit 

—,,       ,.  r  Solution  a 

is  10  amperes.      Ihe  diagrams  tor 

the  third  solution  are  pressure  diagrams,  each  consist- 
ing of  a  horizontal  line  CR  (=100)  units  in  length,  and 
the  impressed  pressure  required  to  pass  10  amperes 
through  the  circuit  is  100  volts.  The  angle  (/>  is  zero.  • 

M 


162  ALTERNATING   CURRENTS. 

b.  The  circuit  consists  of  an  inductive  coil  of  10 
ohms  resistance  and  .01  henry  self-inductance.  The 
phase  and  vector  diagrams  for  the  first  two  solutions 
are  resistance  diagrams,  as  shown  in  the  figure.  The 
impedance  is  shown  by  the  closing  line  of  the  diagram 

R=IO  L=.OI 


R=10 


A'/J- 


27T/L 

=  8. 


CR=ioo iA' 


27T/LC 

=  80 


Sflr 

Solution  6 


80 


100  A, 

to  be  12.8  ohms,  whence  it  is  seen  that  7.81  amperes 
will  flow  through  the  circuit  when  the  impressed  press- 
ure is  100  volts.  The  third  solution  shows  that  the 
impressed  pressure  required  to  pass  10  amperes  through 
the  circuit  is  128  volts.  The  angle  $  is  38°  40'. 


METHODS   OF    SOLVING   PROBLEMS. 


I63 


R=5 


L-.01 


c.  The  circuit  consists  of  a  non-inductive  coil  of  5 
ohms  in  series  with  an  inductive  coil  of  5  ohms  and  .01 
henry.  The  total  phase  and 
vector  diagrams  for  this  are 
exactly  the  same  as  in  ex- 
ample b.  The  vector  dia- 
gram may  be  laid  off  as 
shown  in  the  figure.  This 
shows  that  the  pressure  im- 
pressed upon  the  circuit  as 
a  whole  is  not  equal  to  the 
algebraic  sum  of  the  press- 


Solution  c 


L-.oi 


ures  measured  between  the 
terminals  of  the  parts  of 
the  circuits,  but  it  is  still  equal  to  their  vector  sum. 

d.  The  circuit  consists  of  an  inductance  of  .01  henry. 
The  phase  and  vector  diagrams  for  the  first  two  solu- 
tions are  each  resist- 
ance diagrams  consist- 
ing of  a  vertical  line 
27r/Z(=8)  units  in 
length.  The  impe- 
dance of  the  circuit 
is  8  ohms,  and  the 
current  which  flows 
through  the  circuit 
under  an  impressed 
pressure  of  100  volts 
is  12.5  amperes.  The 
diagrams  for  the  third 
solution  are  pressure  diagrams,  each  consisting  of  a 


=  8 


=  90 


27T/LC 

=  80 


80 


=  90° 


Solution  d 


164 


ALTERNATING   CURRENTS. 


vertical  line  27rfLC(=8d)  units  in  length,  and  the  im- 
pressed pressure  required  to  pass  10  amperes  through 
the  circuit  is  80  volts.  The  angle  <£  is  90°,  and  the  cur- 
rent therefore  lags  90°  behind  the  impressed  pressure. 


o 


The  effect  of  a  condenser  placed  in  series  in  a  circuit 
may  be  shown  by  diagrams  which  are  very  similar  to 
those  relating  to  inductive  circuits.  The  charging  cur- 
rent of  the  condenser  has  such  a  phase  position  and 
magnitude,  that  its  effect  on  the  total  current  flowing 

in    the    circuit    is    the 

A"  same  as  the  effect  of 

an     electric     pressure 

which  is  equal  to — 

27T/T 

and  is  90°  in  advance 

.,    of  the  circuit  current. 
^A 

This  may  be  called  the 
Condenser  Pressure  (see 
Sect.  31).  The  phase 
diagram  is  like  that 
shown  in  Fig.  71.  The 
A'"  pressure  impressed  on 
the  circuit  when  cur- 
rent C  flows,  must 
consist  of  two  compo- 
nents: (i)  the  press- 
ure required  to  pass 
the  current  through 
the  resistance  of  the 
circuit ;  (2)  the  press- 


Fig.  71 


METHODS    OF    SOLVING   PROBLEMS. 


I65 


tire  required  to  balance  the  condenser  pressure.     The 
active  pressure  OA'  in  the  figure  is  equal  to  CR,  and 


the  condenser  pressure  OA"  is  equal  to 


C 


27T/S 


and    is 


90°  in  advance  of  the  active  pressure.  The  capacity  com- 
ponent of  the  impressed  pressure  is  required  to  balance 
,  and  is  therefore  equal  and  opposite  to  OA" .  An 

27T/S 

arrowhead  is  therefore  placed  on  OA"  to  show  that  in 
the  vector  diagram  its  direction  must  be  taken  from  A" 
to  O,  instead  of  from  O  outwards.  The  vector  diagram 
is  then  as  shown  in  Fig.  71  (compare  with  Fig.  76). 

EXAMPLES.  —The  following  examples  are  to  be  solved 
for  the  same  constants  as  before,  the  frequency  being 
taken  as  127^. 


S=ioo 


Circuits  containing  Resistance  and  Capacity. 

e.  The  circuit  contains  simply  a  condenser  having  a 
capacity  of  100  microfarads  (  =  .000100 farad).  The  phase 
and  vector  diagrams 
for  the  first  two  solu- 
tions are  each  resist- 
ance diagrams  consist- 
ing of  a  vertical  line  A' 
-r  (=  12.5)  units  in 

2TTJS 

length.  .  The  impe- 
dance of  the  circuit  is 
12.5  ohms,  and  the 
current  which  flows 
through  the  circuit, 
when  100  volts  is  im- 


C 

)i_    __/ 

V                  C 

N- 

1 

27T/S 

=  12.5 

12.5 

27T/S 

=  125 

^'=-90° 
125 

}                    / 

^i                 C 

)                   / 

Solution  e 

i66  ALTERNATING   CURRENTS. 

pressed  on  it,  is  8  amperes.  The  diagrams  for  the 
third  solution  are  pressure  diagrams  each  consisting  of 

a  vertical  line  :  — -  (=  125)  units  in  length,   and  the 

2  rJt  TS 

impressed  pressure  required  to  pas's  10  amperes  through 
the  circuit  is  125  volts.  The  angle  $  is  —90°,  that  is, 
the  current  is  90°  in  advance  of  the  impressed  pressure. 
The  lines  composing  the  diagrams  for  this  example  are 
drawn  in  a  direction  which  is  exactly  opposite  to  that  of 
the  lines  in  the  diagrams  of  the  example  d. 


=  ioo 


27T/S 
- 12.5 


12.5 


R  =  IO  A 

Solution  / 

f.  The  circuit  consists  of  a  resistance  of  10  ohms 
and  a  capacity  of  100  microfarads.  The  phase  and  vec- 
tor diagrams  are  shown  in  the  figure.  The  impedance 
of  the  circuit  is  16  ohms  and  the  current  which  flows 
under  an  impressed  pressure  of  100  volts  is  6.25  am- 
peres. The  impressed  pressure  required  to  cause  10 


METHODS   OF   SOLVING   PROBLEMS. 


I67 


amperes  to  flow  through  the  circuit  is  160  volts.     The 
angle  </>  is  —  51°  20'. 

Circuits  containing  Self-inductance  and  Capacity. 

g.    The  circuit  consists  of    a  capacity  of  100  micro- 
farads in  series  with  an  inductance  of  .01  henry.     The 

S  - 100  L  -  .01 


/^~^~~^~^\^~^y X 

J—   0  0    o  TVr 


1               4  MI 

27T/S                                   1 

=  12.5 

1 

8 

1 

!   o, 

—  T  

x 

1 

•&<* 

4.5 

=  8                                                      A; 

Solution  g 

diagrams  are  as  shown.  The  impedance  of  the  circuit 
is  4.5  ohms.  The  current  which  flows  when  the  im- 
pressed pressure  is  100  volts  is  22,2  amperes,  at  which 
time  the  pressure  measured  between  the  terminals  of 
the  condenser  is  277.^  volts  and  that  measured  between 


1 68 


ALTERNATING   CURRENTS. 


the  terminals  of  the  inductance  is  177.6  volts.  The 
impressed  pressure  required  to  pass  10  amperes  through 
the  circuit  is  45  volts.  The  angle  (f>  is  —  90°. 

//.    The  circuit  consists  of  a  capacity  of    125  micro- 
farads in  series  with  an  inductance  of  .015  henry.     The 


-=  .015 


8=135 


A-r 


1 
1 

1 

'lv 

1 
1 
1 

3 

37T/3 

! 

!    A. 
1 
* 

iT90 

A  * 


27T/L 

=  12 


Solution   li 


impedance  of  the  circuit  is  2  ohms  and  the  current 
which  flows  under  100  volts  impressed  pressure  is  50 
amperes,  at  which  time  the  pressure  measured  between 


METHODS    OF    SOLVING   PROBLEMS. 


169 


the  terminals  of  the  condenser  is  500  volts  and  between 
the  terminals  of  the  inductance  is  600  volts.  The  im- 
pressed pressure  required  to  pass  10  amperes  through 
the  circuit  is  20  volts.  With  this  current  flowing,  the 


L=.156 


S=.ioo 


i 

27T/S 


27T/L 

=  12.5 


Solution  i 


pressure  measured  between  the  terminals  of  the  con- 
denser is  100  volts  and  between  the  terminals  of  the 
inductance  is  120  volts.  The  angle  <j>  is  90°. 

i.    The  circuit  consists  of  a  capacity  of   100  micro- 
farads  in    series  with    an    inductance   of    .0156   henry. 


ALTERNATING   CURRENTS. 


R=10,L=.Ol  s=100 


A" 


1 

27T/S 

=12.5 


R  =  10  A 


27T/L 


=  10,  L=.01 


10 


A, 


,14 


TT 
t 


100 


Solution  j 


METHODS    OF    SOLVING   PROBLEMS.  171 

The  phase  and  vector  diagrams  are  shown  in  the  figure. 
In   this   case  -=•  =  2  jrfL  and   the  impedance  of  the 

27T/S 

circuit  is  zero. 

Circuits  containing  Resistance,  Self -inductance, 
and  Capacity. 

j.  The  circuit  consists  of  10  ohms  in  series  with  100 
microfarads  and  .01  henry.  The  impedance  of  the  cir- 
cuit is  shown  by  the  diagram  to  be  1 1  —  ohms.  The 
current  which  flows  through  the  circuit  when  100  volts 
are  impressed  at  its  terminals  is  9.1  amperes.  The 
pressure  required  to  pass  10  amperes  through  the  cir- 
cuit is  no  —  volts.  This  is  the  vector  sum  of  125  and 
128,  which  are  the  pressures  measured  respectively 
between  the  condenser  terminals  and  the  remainder  of 
the  circuit.  The  angle  <£  is  —  24°  14'. 

k.  The  circuit  consists  of  10  ohms  in  series  with  150 
microfarads  and  .01042  henry.  The  diagrams  show  that 
the  impedance  of  the  circuit  is  ten  ohms.  One  hundred 
volts  is  therefore  the  impressed  pressure  that  gives  a 
current  of  10  amperes.  When  10  amperes  flow  in  the 
circuit,  the  pressure  measured  between  the  terminals  of 
the  condenser  is  83.3  volts,  and  that  measured  between 
the  terminals  of  the  remainder  of  the  circuit  is  130  volts. 
The  angle  <£  is  zero. 

51.  Conclusions  in  regard  to  Series  Circuits. — The 
eleven  examples  thus  given  cover  every  fundamental 
arrangement  of  series  circuits  which  may  occur.  An 
examination  of  the  diagrams  and  of  the  principles 
involved  in  their  construction,  makes  the  following 
statements  evident  : 


172 


ALTERNATING    CURRENTS. 


1 

27T/8 

=  8.33 


R  -  10,         L=  .01042      S  -  150 


27T/L 

=  8.33 


•A" 

R=10,  L=.0104 


A' 


10 


R  =  10,  S  =  150 


100 


8.33 


100 


Solution  7c 


METHODS    OF    SOLVING   PROBLEMS.  173 

1.  When    non-inductive    circuits    are    connected    in 
series,  the  total  impressed  pressure  equals  the  sum  of 
the  pressures  measured  between  the  terminals  of  the 
individual  parts,  and  the  total  resistance  of  the  circuit  is 
equal  to  the  sum  of  the  resistances  of    the  individual 
parts.  , 

2.  When    inductive  circuits   of  equal  time  constants 
are  connected  in  series,   the   total   impressed   pressure 
equals  the   sum   of   the   pressures  upon   the  individual 
parts,  measured  between  their  individual  terminals,  and 
the  total  impedance  of  the  circuit  is  equal  to  the  sum  of 
the  individual  impedances. 

3.  When  inductive  and  non-inductive  circuits  are  con- 
nected   in   series   with   each    other,   or   when   inductive 
circuits    of    unequal  time   constants   are    connected    in 
series,  the  total  impressed  pressure    equals    the  vector 
sum,  which  is  always  less  than  the  algebraic  sum,  of  the 
pressures  measured  between  the  terminals  of  the  indi- 
vidual parts,  and  the  individual  pressures  are  each  less 
than  the   total    impressed   pressure.     The  total  impe- 
dance of  the  circuit   is  equal  to  the  vector  sum  of  the 
individual  impedances,  each  of  which  is  less  than  the 
total. 

4.  When  condensers  are  connected  in  series  by  con- 
ductors  of   negligible    resistance,   the    total    impressed 
pressure    equals    the    sum    of   the  pressures    measured 
across    the    individual    condensers,   and   the   total    im- 
pedance   of   the    circuit    is    equal    to    the    sum    of   the 
impedances  of  the  individual  condensers. 

5.  When    condensers   are    connected    in    series    with 
non-inductive    resistances^  the    total  impressed  pressure 


ALTERNATING   CURRENTS. 

equals  the  vector  sum,  which  is  always  less  than  the 
algebraic  sum,  of  the  pressures  measured  between  the 
terminals  of  the  individual  parts  of  the  circuit,  and 
the  individual  pressures  are  each  less  than  the  total  im- 
pressed pressure.  The  total  impedance  of  the  circuit 
is  equal  to  the  vector  sum  of  the  individual  impedances, 
each  of  which  is  less  than  the  total. 

6.  When  condensers  are  connected  in  series  with 
inductive  resistances,  the  total  impressed  pressure  equals 
the  vector  sum,  which  is  always  less  than  the  algebraic 
sum,  of  the  pressures  measured  between  the  terminals 
of  the  individual  parts  of  the  circuit.  Since  the  effects  of 
capacity  and  self-inductance  respectively  cause  the  angle 
$  to  become  negative  and  positive,  the  individual  press- 
tires  may  be  either  greater  or  less  than  the  total  impressed 
pressure,  depending  upon  the  relation  between  the  vari- 
ous resistances,  capacities,  and  inductances  in  the  cir- 
cuit. The  total  impedance  of  the  circuit  is  equal  to  the 
vector  sum  of  the  individual  impedances,  each  of  which 
may  be  either  greater  or  less  than  the  total  impedance. 

The  third,  fifth,  and  sixth  paragraphs  above  make  the 
following  proposition  evident :  When  in  series  circuits, 
the  angles  taken  between  the  phases  of  the  current  and 
the  individual  pressures,  measured  at  the  terminals  of  the 
parts  of  the  circuit,  are  all  either  positive  or  negative, 
the  total  impressed  pressure  is  always  greater  than  any 
of  the  individual  or  partial  pressures.  When  the  angles 
taken  between  the  phases  of  the  current  and  the  partial 
pressures  are  in  part  positive  and  in  part  negative,  some 
or  all  of  the  partial  pressures  may  be  greater  than  the 
total  impressed  pressure. 


METHODS    OF   SOLVING   PROBLEMS.  175 

52.  Parallel  Circuits.  —  Second  Class.  The  graphical 
treatment  of  problems  relating  to  parallel  circuits  is  en- 
tirely analogous  to  that  given  for  series  circuits.  As  the 
simplest  cases  of  parallel  circuits  are  those  in  which  the 
same  electrical  pressure  is  impressed  upon  all  the  parts 
of  the  circuit,  these  will  be  treated  first.  In  this  class 
the  same  general  operations  are  used  in  solving  prob- 
lems as  in  the  first  class,  but  alternating  currents  and 
proportional  conductivities  are  dealt  with  instead  of 
alternating  pressures  and  proportional  resistances.  Sup- 
pose a  circuit  made  up  of  two  branches  in  parallel,  each 
with  a  known  resistance  and  reactance,  and  it  is  desired 
to  know  what  impressed  pressure  with  a  frequency  f  is 
required  to  pass  through  it  a  certain  current  C.  In  this 
case,  the  total  current  is  made  up  of  two  components, 
each  of  which  flows  through  one  of  the  branches  and  is 
inversely  proportional  to  the  impedance  of  the  branch, 
and  the  phase  of  which  has  an  angular  retardation  with 
respect  to  that  of  the  impressed  pressure  which  depends 
upon  the  time  constant  of  the  branch.  The  total  cur- 
rent, which  is  inversely  proportional  to  the  equivalent 
impedance  of  the  parallel  circuit,  is  equal  in  magnitude 
and  position  to  the  resultant  of  the  branch  currents. 
The  condition  is  represented  by  Fig.  72,  in  which  OA 
and  OA'  are  the  currents  in  the  two  branches  respec- 
tively, <p  and  <£/  being  their  respective  lag  angles.  The 
relative  phase  position  of  the  impressed  pressure  is 
taken  on  the  horizontal  line.  Then  the  resultant  or 
total  current  in  the  circuit  is  represented  in  magnitude 

r? 

and  phase  by  OA".     OA  is  equal  to  — ,  OA'  is  equal  to 

* 


1 76  ALTERNATING   CURRENTS. 

A' 


Fig.  72 


— x 


\<v 


Pig.  73 


METHODS   OF    SOLVING   PROBLEMS.  177 

27  27 

—  ,  and  OA"  is  equal  to  —  ,  where  E  is  the  pressure  im- 
^2  * 

pressed  on  the  circuits,  and  the  denominators   of  the 

fractions  are  the  respective  impedances  of  the  branches 
and  of  the  total  circuit.  It  is  therefore  evident  that  the 
reciprocal  of  the  equivalent  impedance  (=  the  equiva- 
lent apparent  conductivity)  on  a  parallel  circuit  may 
be  at  once  derived  from  the  apparent  conductivities  of 


=  100 


1=^=6.06  OHMS. 

lo.o 

Solution  6 
Pig.  74 

the  branches,  by  taking  their  vector  sum,  as  is  shown 
in  Fig.  73.  The  equivalent  apparent  conductivity,  or 
the  equivalent  impedance  of  a  circuit  being  known,  the 
pressure  required  to  pass  a  given  current  through  it,  or 
the  current  flowing  under  a  given  impressed  pressure, 
may  be  at  once  derived.  In  dealing  with  series  cir- 
cuits, the  phase  of  the  current  (or  active  pressure)  has 
been  assumed  to  be  along  the  horizontal  line,  or  line  of 
reference.  In  dealing  with  parallel  circuits,  it  is  more 


1/8         ALTERNATING  CURRENTS. 

convenient  to  assume  the  phase  of  the  impressed  press- 
ure (or  apparent  conductivity)  for  the  reference  phase. 
It  must  always  be  remembered,  however,  that  angles 
of  lag  are  measured  from  lines  representing  current  to 
lines  representing  pressure.  Thus,  in  Fig.  74,  the 
angle  </>  is  positive  because  the  current  lags  behind  the 
pressure. 

EXAMPLES.  —  In  the  following  examples  it  is  desired 
to  find  for  each  of  the  given  circuits  :  (i)  the  equivalent 
impedance  of  the  circuit ;  (2)  the  current  which  flows 
through  the  circuit  when  the  impressed  pressure  is  100 
volts ;  (3)  the  impressed  pressure  which  is  required  to 
pass  10  amperes  through  the  circuit ;  (4)  the  angle  by 
which  the  total  current  lags  behind  the  impressed  press- 
ure. The  frequency  is  taken  as  in  the  examples  of  the 
first  class  to  be  I27-|-,  whence  2  73/is  equal  to  800. 

Circuits  containing  Resistance  and  Self -inductance. 

a.  The  circuit  consists  of  two  non-reactive*  branches 
in  parallel,  one  having  20  ohms  resistance  and  the  other 
10  ohms  resistance.  The  phase  diagram  for  the  solu- 
tions, using  the  apparent  conductivities  of  the  circuits 
as  a  basis  of  work,  is  two  horizontal  lines  superposed,  of 
lengths  respectively  .05  and  .10  unit.  The  vector  dia- 

*  The  terms  inductive,  capacity,  and  reactive  circuit,  will  hereafter  be 
used  with  the  following  significations  :  an  inductive  circuit  is  one  contain- 
ing inductance,  but  not  capacity  ;  a  capacity  or  condenser  circtiit  is  one 
containing  capacity,  but  not  inductance;  a  reactive  circuit  is  one  contain- 
ing either  inductance  or  capacity  or  both  inductance  and  capacity.  A 
non-reactive  circuit  is,  therefore,  one  which  contains  neither  inductance 
nor  capacity,  that  is,  one  which  contains  a  plain  resistance  only. 


METHODS   OF    SOLVING   PROBLEMS. 


179 


gram  is  given  by  drawing  these  consecutively,  and 
the  equivalent  apparent  conductivity  of  the  circuit  is 
in  the  figure.  The  equivalent  impedance  of  the 


A' 


.10 


.05 


~  .15 

Solution  a 

circuit  is  therefore  6.67  ohms.  The  current  flowing 
under  100  volts  pressure  is  15  amperes,  and  the  press- 
ure required  to  pass  10  amperes  through  the  circuit  is 
66.7  volts. 

b.  The  circuit  consists  of  a  non-reactive  branch  of  10 
ohms  and  an  inductive  branch  of  .01  henry.  The  phase 
diagram  consists  of  two  lines  at  right  angles  (one  being 
horizontal),  since  the  current  in  the  non-reactive  branch 
is  in  phase  with  the  impressed  pressure,  and  that  in  the 
inductive  branch  lags  90°  behind  the  impressed  pressure. 
The  lengths  of  the  lines  are  respectively  -  -  (=.io) 


units  and 


2  7T/L 


(=  .125)  units.     The  vector  diagram  is 


shown.  The  equivalent  conductivity  of  the  circuit  is 
.16  and  the  impedance  is  6.25  ohms.  The  current 
flowing  under  an  impressed  pressure  of  100  volts  is  16 


i8o 


ALTERNATING  CURRENTS. 


amperes,  and  it  requires  62.5  volts  to  cause  10  amperes 
to  flow.     The  angle  </>  is  51°  20'. 


!-=  10 


.10 


27T/L 
=.125 


.125 


Solution  b 


c.  The  circuit  consists  of  a  non-reactive  branch  of  10 
ohms  and  an  inductive  branch  having  a  resistance  of 
10  ohms  and  an  inductance  of  .01  henry.  The  impe- 
dance and  the  angle  of  lag  for  the  inductive  branch  are 
found  by  the  method  given  under  Series  Circuits  :  First 
Class,  and  the  conductivity  of  the  branch  is  laid  off 
in  the  phase  diagram  on  a  line  making  with  the  hori- 
zontal axis  an  angle  equal  to  the  angle  of  lag  taken 
backwards.  This  is  line  OA"  in  the  diagram.  The  line 
OA'  represents  the  conductivity  of,  and  the  relative 
phase  of  current  in,  the  non-reactive  branch.  The 
length  and  direction  of  the  line  OA2  in  the  vector 


METHODS   OF    SOLVING   PROBLEMS. 


181 


diagram  shows  the  value  of  the  equivalent  or  joint  con- 
ductivity of  the  circuit,  and  the  angle  by  which  the  phase 
of  the  main  current  lags  behind  the  phase  of  the  im- 
pressed pressure.  The  joint  conductivity  of  the  circuit 


10 


is  .168  and  the  joint  impedance  5.95  ohms.  The  current 
flowing  under  an  impressed  pressure  of  100  volts  is  16.8 
amperes,  and  the  pressure  required  to  pass  10  amperes 
through  the  circuit  is  59.5  volts.  The  angle  </>  is  16°  52'. 
d.  The  circuit  consists  of  two  inductive  branches  of 


182 


ALTERNATING   CURRENTS. 


respectively  .01  and  .0125  henry.  The  diagrams  con- 
sist of  vertical  lines  as  shown.  The  conductivity  is 
.225  and  the  impedance  is  4.44  ohms.  The  current 


L  2  =.0125 


A" 
A'- 


°i 


Af 


Solution  d 

flowing  under  100  volts  pressure  is  22.5  amperes,  and 
it  requires  44.4  volts  to  cause  10  amperes  to  flow.  The 
angle  of  lag  is  90°. 

e.  The  circuit  consists  of  two  reactive  branches  of 
respectively  .005  henry  and  10  ohms,  and  .0125  henry 
and  8  ohms.  The  diagrams  are  as  shown.  The  conduc- 
tivity of  the  circuit  is  .165,  and  the  impedance  is  6.06 
ohms.  The  current  flowing  under  100  volts  pressure  is 


METHODS    OF    SOLVING   PROBLEMS. 


183 


16.5  amperes,  and  the  pressure  required  to  pass  10 
amperes  through  the  circuit  is  60.6  volts.  The  angle  </> 
is  35°  16'. 

The  five  preceding  examples  cover  all  the  fundamen- 
tal combinations  of  resistance  and  inductance  in  parallel 


Solution  e 

circuits.  The  following  four  in  like  manner  cover  the 
combinations  of  resistance  and  capacity.  The  solutions 
in  the  two  cases  are  entirely  similar,  but  the  lag 
angles  become  negative  on  account  of  the  influence 
of  capacities. 

Circuits  containing  Resistance  and  Capacity. 

f.  When  two  or  more  condensers  are  connected  in 
parallel  by  wires  of  negligible  resistance,  they  evidently 
act  upon  the  circuit  exactly  as  though  it  contained 
one  condenser  with  a  capacity  equal  to  the  combined 


1 84 


ALTERNATING    CURRENTS. 


capacity    of    those    in    parallel.     The    impedance    of    a 


condenser  is   equal   to 


2  ir/s 


,  and  its  apparent  conduc- 


tivity to  2  TT/S.     The  apparent   conductivity  of  several 
condensers  in  parallel  is  therefore  evidently 

2irf(s1  +  s2+  etc.) 

g.  The  circuit  consists  of  a  non-reactive  branch  of 
10  ohms  and  a  capacity  branch  of  100  microfarads.  The 
diagrams  are  as  shown.  The  conductivity  of  the  circuit 

R-io 


=  ioo 


27T/S 


-=.10 


Pi 

Solution  g 


is.  128  and  its  impedance  is  7.82  ohms.  The  current 
flowing  under  a  pressure  of  100  volts  is  12.8  amperes, 
and  the  pressure  required  to  pass  10  amperes  through 
the  circuit  is  78.2  volts.  The  angle  <£  is  —  38°  40'. 

h.  The  circuit  consists  of  a  non-reactive  branch  of 
10  ohms,  and  a  reactive  branch  of  10  ohms  and  100 
microfarads.  The  conductivity  of  the  circuit  is  shown 


METHODS    OK   SOLVING   PROBLEMS. 


I85 


to  be  .147  and  the  impedance  is  6.8  ohms.     The  cur- 
rent flowing  under  100  volts  pressure  is   14.7  amperes, 


R=10 


R=io,  s=ioo 


and  the  pressure  required  to  pass  10  amperes  through 
the  circuit  is  68  volts.     The  angle  of  lag  is  —  19°  20'. 

i.  The  circuit  consists  of  two  reactive  branches, 
respectively  of  10  ohms  and  100  microfarads,  and  of 
20  ohms  and  250  microfarads.  The  conductivity  of  the 
circuit  is  shown  to  be  .105  and  the  impedance  is  9.5 
ohms.  The  current  flowing  under  a  pressure  of  100 
volts  is  10.5  amperes,  and  the  pressure  required  to  pass 
10  amperes  through  the  circuit  is  95  volts.  The  angle 
<£  is  —  35°  10'. 


1 86 


ALTERNATING   CURRENTS. 


R=10,          s  =  100 


Solution  i 

The  following  examples  cover  the  fundamental  com 
binations  of  capacities  and  inductances. 

j.  The  circuit  consists  of  two  reactive  branches,  re 
spectively  of  5  ohms  and  .005  henry,  and  of  10  ohms 
and  100  microfarads.  The  conductivity  of  the  circuit 
is  shown  to  be  .168  and  the  impedance  is  5.95  ohms 
The  current  flowing  under  a  pressure  of  100  volts  is 


METHODS   OF    SOLVING   PROBLEMS.  187 

1 6. 8  amperes,  and  the  pressure  required  to  cause  a 
current  of  10  amperes  is  59.5  volts.  The  angle  </>  is 
1 6°  50'. 

R=5,  L=.005 


k.  The  circuit  consists  of  two  reactive  branches, 
respectively  of  10  ohms  and  .0156  henry,  and  of  5  ohms 
and  200  microfarads.  The  conductivity  is  shown  to  be 
.127  and  the  impedance  of  the  circuit  is  7.87  ohms. 


1 88 


ALTERNATING   CURRENTS. 


The  current  flowing  under  a  pressure  of  100  volts  is 
12.7  amperes,  and  78.7  volts  are  required  to  pass  10 
amperes  through  the  circuit.  The  angle  0  is  —  22°  37'. 


/.  The  circuit  consists  of  two  reactive  branches  of 
respectively  10  ohms  and  .01042  henry,  and  10  ohms 
and  150  microfarads.  The  diagrams  show  that  the  im- 


METHODS   OF    SOLVING    PROBLEMS. 


189 


pedance  of  the  circuit  is  8.47  ohms  and  the  angle  <f>  is 
equal  to  zero. 

R=10,     L  =  .01042 


=  150 


x 


Solution  I 

m.  The  circuit  .consists  of  two  reactive  branches  re- 
spectively of  10  ohms  and  .01  henry,  and  of  100  micro- 
farads. The  diagrams  show  the  joint  impedance  to  be 
14.5  ohms.  The  impedances  of  the  branches  are  re- 
spectively 12.8  and  12.5,  so  that  when  the  impressed 
pressure  is  100  volts,  6.9  amperes  flow  in  the  main 
circuit,  while  8  and  7.8  amperes  respectively  flow  in 
the  branches.  The  angle  $  is  —  27°  10'. 

n.    The  circuit  consists  of  two  reactive  branches  re- 


igo 


ALTERNATING   CURRENTS. 


spectively  of  .01  henry,  and  of  10  ohms  and  100  micro- 
farads. The  diagrams  show  the  impedance  to  be  n.6 
ohms.  The  impedances  of  the  branches  are  respec- 
tively 8  and  16,  so  that  when  100  volts  pressure  is 
impressed  upon  the  circuit  8.6  amperes  flow  in  the 


R-10,  L-.M 


main   leads,  while   12.5  and  6.25  amperes  flow  respec- 
tively in  the  two  branches.     The  angle  $  is  62°  53'. 

o.  The  circuit  consists  of  two  reactive  branches  re- 
spectively of  .01  henry  and  of  100  microfarads.  The 
impedance  of  the  circuit  is  22.2  and  the  impedances 
of  the  branches  are  respectively  8  and  12.5  ohms.  When 
the  impressed  pressure  is  100  volts,  the  main  current  is 


METHODS   OF    SOLVING   PROBLEMS. 


191 


4.5   amperes  and  that  in  the  branches  is   12.5   and  8 
amperes.     The  angle  <f>  is  90°. 

/.    The  circuit  consists  of  two  reactive  branches  of 
respectively  .01042  henry  and  150  microfarads.    The  dia- 
L=.oi 


R=10,  8=100 


A 


10 


12.5 


L— X 


Pit— 7 -> 


=  62  53 
.125 


Aa 


.125 


Solution  n 

grams  show  that  the  two  branch  currents  are  in  exact 
opposition  and  of  equal  value  and  that  the  joint  con- 
ductivity is  zero,  so  that  the  main  current  is  zero. 
When  the  impressed  pressure  is  100  volts  the  branch 
currents  are  each  12  amperes,  and  when  10  amperes 
flow  in  each  branch  the  pressure  is  83.3  volts. 


192  ALTERNATING   CURRENTS. 

AV  L=.0l 

pnnr 

.08 


o-  - 


A"- 


.135 


\ 


8=100 


4 


Solution  o 


.13 


.12 


"X 


L=.  01042 


=  150 


Solution 


METHODS    OF   SOLVING   PROBLEMS.  193 

53.  Conclusions  in  Regard  to  Parallel  Circuits.  — Sec- 
ond Class.  The  sixteen  examples  just  presented  cover 
every  fundamental  arrangement  of  simple  parallel  cir- 
cuits. An  examination  of  the  diagrams  and  the  prin- 
ciples involved  in  their  construction  makes  evident  the 
following  statements,  which  are  in  many  respects  anal- 
ogous to  those  given  as  applying  to  series  circuits : 

1.  When  non-reactive  circuits  are  connected  in  par- 
allel,  the  total   current   equals    the    algebraic    sum    of 
the  currents  in  the  branches,  and  the  joint  conductivity 
of   the  circuit   is    equal  to  the  algebraic  sum   of    the 
branch  conductivities. 

2.  When  inductive   circuits   of  equal  time  constants 
are  connected  in  parallel,  the  total  current  equals  the 
algebraic  sum  of  the  currents  in  the  branches,  and  the 
joint  conductivity  of  the  circuit  is  equal  to  the  alge- 
braic sum  of  the  branch  conductivities. 

3.  When  inductive  and  non-reactive  circuits  are  con- 
nected in  parallel  with  each  other,  or  when  inductive  cir- 
cuits of  unequal  time  constants  are  connected  in  parallel, 
the  total  current  is  equal  to  the  vector  sum,  which  is 
always  less  than  the  algebraic  sum,  of  the  branch  cur- 
rents, and  the  individual  branch  currents  are  each  smaller 
than  the  total  current.      The  joint  conductivity  of  the 
circuit  is  equal  to  the  vector  sum  of  the  branch  con- 
ductivities, each  of  which  is  less  than  the  joint  total. 

4.  When    condensers   are    connected    in    parallel    by 
wires  of  negligible  resistance,  the  total  current  equals 
the  algebraic  sum  of  the  branch  currents,  and  the  joint 
conductivity  equals  the  algebraic  sum  of   the  branch 
conductivities. 


194  ALTERNATING   CURRENTS. 

5.  When  condensers  are   connected   in   parallel  with 
non-reactive  resistances,  the  total  current  equals  the  vec- 
tor sum,  which  is  always  less  than  the  algebraic  sum, 
of  the  branch  currents,  and  the  individual  branch  cur- 
rents are  each  smaller  than  the  total  current.     The  joint 
conductivity  of  the  circuit  equals  the  vector  sum  of  the 
branch  conductivities,  each  of  which  is  smaller  than  the 
joint  total. 

6.  When  condensers  are   connected   in  parallel  with 
inductive  circuits,   the  total  current   equals   the  vector 
sum,  which  is  always  less  than  the  algebraic  sum,  of 
the    currents    in    the    branches.     Since  the  effects   of 
capacity  and  of  self-inductance  respectively  cause  the 
angle  <f>  to  become  negative  and  positive,  the  individual 
branch  currents  may  be  either  greater  or  less  than  the 
main  or  total  current,  depending  upon  the  relation  be- 
tween the  various  capacities  and  inductances  in  the  cir- 
cuit.    The  joint  conductivity  of  the  circuit  equals  the 
vector  sum  of  the  branch  conductivities,  each  of  which 
may  be  either  greater  or  less  than  the  joint  conductivity. 

The  third,  fifth,  and  sixth  paragraphs  make  evident 
this  proposition,  which  is  similar  to  that  given  for  series 
circuits  (Sect.  51):  When  in  parallel  circuits  the  cur- 
rents in  the  branches  are  all  either  lagging  or  leading 
with  respect  to  the  impressed  pressure,  the  total  or 
main  current  is  always  greater  than  the  current  in 
any  one  of  the  branches.  When  the  currents  in  part 
of  the  branches  lead  the  impressed  pressure  and  in 
other  branches  lag  behind  the  pressure,  some  or  all  of 
the  branch  currents  may  be  greater  than  the  total  or  main 
current.  It  is  even  theoretically  possible  for  the  angles 


METHODS   OF    SOLVING   PROBLEMS.  195 

have  such  •  a  relation  that  a  large  current  may  flozv  in 
the  branches  while  the  main  current  is  zero. 

54.  With  the  methods  thus  set  forth  it  is  possible  to 
solve  any  problem  which  may  arise  in  regard  to  the  im- 
pedance presented  by  any  circuit  to  the  flow  of  a  sinu- 
soidal current.  When  the  current  is  not  sinusoidal  the 
deductions  do  not  strictly  apply,  but  for  the  alternating 
currents  which  are  commonly  found  in  practice  the 
approximation  of  the  deductions  to  the  facts  is  fairly 
close.*  In  every  case  it  is  assumed  that  the  parts  of 
the  circuits  have  no  appreciable  mutual  magnetic  effect. 
If  the  parts  are  mutually  inductive,  the  solutions  be- 
come entirely  different  and  much  more  complicated. 

The  solutions  for  parallel  circuits  may  be  made  by 
another  method  in  which  pressures  and  impedances  are 
involved.  This  method  may  be  best  exemplified  by 
illustrations.  Suppose,  for  instance,  it  is  desired  to  find 
the  joint  impedance  of  the  branched  circuit  in  example  e 
(Sect.  52).  It  may  be  assumed  that  an  impressed  press- 
ure of  100  volts  acts  on  the  circuit.  Upon  a  line,  OX, 
representing  this  pressure  (Fig.  74)  is  drawn  a  semi- 
circle. From  O  draw  the  line  OA  making  a  lag  angle  of 

(/>!  with  OX,  where  tan  fa  =      ^    *  =  .4.     Then  OA  is 

K1 

equal  to  C1R1  and  XA  is  equal  to  2irfLlCl  since  the 

angle  at  A  is  a  right  angle.     The  current  in  this  branch 

/")  A 
when   the   impressed   pressure   is  equal   to  OX,  is  -  — , 

\ 

and  this  may  be  laid  off  from  O  to  B.     The  current  in 


*  Compare   Bedell   on   hedgehog  transformer  with  condenser,    Trans, 
Amer.  Inst.  E.  E.,  Vol.  10,  p.  515. 


196          ALTERNATING  CURRENTS. 

the  second  branch  is  given  by  laying  off  the  direction 
of  the  line  OA'  so  that  it  makes  a  lag  angle  of  </>2  with 
OX,  where  tan  <£2  =  SSLj  =  l  2^  The  current  in  the 

A2 

second  branch  is  equal  to  OA'  divided  by  R^  and  when 
laid  off  from  O  gives  OB1.  The  total  current  in  the 


X 


__  6.06  OHMS. 

Solution  e 
Fig.  74 


circuit  is  the  resultant  of  OB  and  OB1,  or  OB'1.     Its 
value  in  amperes  is  16.5.     The  impedance  of  the  circuit 


is  then       =        -  =  ~  =  6.06  ohms.    The  angle  of  lag 

I  O  .  S 


is  <J,  =  tan-' ^    =  35°  16'.      Figures  75,  76,  77,  78, 
\l//2    /  . 

79,  and  80  give  the  solutions  by  the  same  method  for 
examples  b,  c,  dt  g,  h,j  of  Section  52.  These  show  the 
application  of  the  method  fully.* 

55.  Series  and  Parallel  Circuits  Combined.  —  Third 
Class.  Where  series  and  parallel  circuits  are  combined 
the  fundamental  solutions  already  given  apply  directly, 

*  Compare  Bedell  and  Crehore,  Alternating  Currents,  p.  292;  Loppe 
et  Bouquet,  Courants  Alternatifs  Industriels,  p.  ill. 


METHODS    OF   SOLVING   PROBLEMS. 


5'95  OHMS- 
Solution    c 
Fig.  76 


198  ALTERNATING   CURRENTS. 

O  100 


TTT 


Fig.  77 


,  =  ^=7.82 
Solution  Q 


B' 
Fig.  78 


100 


METHODS    OF    SOLVING   PROBLEMS.  199 


Solution  h 
Fig.  79 


Solution  j 
Fig.  80 


2OO 


ALTERNATING    CURRENTS. 


and  it  simply  requires  experience  to  acquire  consider- 
able facility  in  the  solutions  relating  to  the  most  com- 
plicated circuits.  Several  examples  are  given  below  to 


A3 


10 


O  10  A 

1=18.8 

Solution  a 

indicate  the  general  procedure.  In  these  examples  it  is 
desired  to  determine  as  before  :  (i)  the  total  impedance 
of  the  circuit ;  (2)  the  total  current  flowing  under  a 
pressure  of  100 volts;  (3)  the  pressure  required  to  cause 


METHODS    OF    SOLVING   PROBLEMS.  2OI 

10  amperes  to  flow ;  and  (4)  the  lag  angle  between  the 
total  current  and  the  impressed  pressure.  The  frequency 
is  taken  as  I2/J-. 

a.  The  circuit  consists  of  an  inductive  coil  of  10  ohms 
and  .01  henry  in  series  with  a  branched  circuit  similar 
to  e,  Sect.  52.     We  know  that  the  parallel  part  of  the 
circuit  has  an  impedance  of  6.06  ohms,  and  that  the  lag 
angle  is  35°  i6f.     Therefore  OAf  is  laid  off  in  the  phase 
diagram  6.06  units  in  length,  and  making  the  proper 
angle  with  the  horizontal.     The  line  OA"  is  then  laid 
off  horizontally  R% (=  10)  units  long,  and  OA'"  is  laid 
off  vertically  2  7rfL3(=  8)  units  long.      In  the  vector  dia- 
gram O1A1  is  equal  and  parallel  to  OA1 ',  A±A%  to  OA", 
and  A2AS  to  OA'".     The  length  of  the  line  O^3  gives 
the  impedance  of   the   circuit,  which   is   equal    to   18.8 
ohms.     The  current  which  flows  under   a   pressure  of 
100  volts  is  5.32  amperes,  and  it  requires   188  volts  to 
cause    10  amperes   to  flow   through    the   circuit.     The 
angle  of  lag,  </>,   is  the  angle  A^O^X,  and  is  equal  to 

37°  34'. 

b.  The  circuit  consists  of  an  inductance  of  .01  henry 
in  series  with  a  branched  circuit  having  two  branches 
containing   respectively  40  ohms  and   100  microfarads. 
The  joint  impedance  of  the  branched  part  of  the  circuit 
is  first  found  in  the  usual  manner.     This  is  11.9  ohms, 
and  the  lag  is  —  72°  40'.     In  the  phase  diagram,  OA'  is 
therefore  laid  off  equal  to  11.9  and  making  a  lag  angle 
of  -72°  40',  and  OA"  is  laid   off   2w/£(=8)  units  in 
length  and  making  a  lag  angle  of  90°.     Laying  off  the  vec- 
tor diagram  gives  <9^42  equal  to  4.68  and  making  a  lag 
angle  of  —  43°  40'.     The  current  flowing  under  a  press- 


202  ALTERNATING   CURRENTS. 

A'  40^  R 


O  .025    A" 


1  =  13.5 

Solution    6j 


METHODS    OF   SOLVING    PROBLEMS.  203 

lire  of  100  volts  is  21.4  amperes,  and  the  pressure  required 
to  cause  10  amperes  to  flow  is  46.8  volts.  When  10 
amperes  flow  in  the  main  circuit,  the  pressure  at  the 
terminals  of  the  branched  circuit  is  119  volts,  and  the 
currents  which  flow  through  the  resistance  and  the  con- 
denser are  respectively  3  amperes  and  9.5  amperes.  The 
pressure  across  the  inductance  Z3  is  then  80  volts. 

b^.  If  the  frequency  in  the  preceding  example  is  cut 
down  to  80,  the  relations  are  materially  changed.  The 
impedance  of  the  branched  circuit  becomes  17.8  ohms, 
and  the  lag  angle  in  it  is  —  63°  32'.  The  phase  dia- 
gram, therefore,  is  as  shown.  From  the  vector  diagram 
it  is  seen  that  the  joint  impedance  of  the  whole  circuit 
is  13.5  ohms,  and  the  total  current  is  54°  c/  ahead  of  the 
impressed  pressure.  The  total  current  flowing  when 
the  impressed  pressure  is  100  volts  is  7.4  amperes,  and 
it  requires  135  volts  to  cause  10  amperes  to  flow.  When 
10  amperes  are  flowing,  the  pressure  at  the  terminals 
of  the  branched  circuit  is  178  volts,  and  the  currents 
which  flow  through  the  resistance  and  the  condenser  are 
4.45  and  8.9  amperes  respectively,  while  the  pressure 
across  the  inductance  Lz  is  50  volts.  To  maintain  a 
pressure  of  100  volts  at  the  terminals  of  the  divided  cir- 
cuit requires  an  impressed  pressure  of  76  volts.  With 
this  pressure  5.6  amperes  flow  through  the  circuit. 

c.  The  circuit  consists  of  a  combination  as  shown  in 
the  figure  on  page  204.  The  resistances  of  the  branches 
of  the  circuit  are  R1  =  5  ohms,  R^  =  10  ohms,  7?3  =  8 
ohms;  the  inductances  are  Ll  =  .005  henry,  L2  =  .oi 
henry,  Z3  =  .0125  henry  ;  and  the  capacities  are  sl=  150 
microfarads,  s2  =  100  microfarads,  ss  =  125  microfarads. 


2O4 


ALTERNATING   CURRENTS. 


oo  CD 


"(CD 


CD 


QQ 


METHODS   OF   SOLVING   PROBLEMS. 


205 


The  diagrams  show  the  impedance  of  the  branched 
part  of  the  circuit  to  be  4.74  ohms,  and  its  lag  angle  is 
—  10°  12'.  From  the  complete  diagrams  it  is  seen  that 
the  joint  impedance  of  the  whole  circuit  is  10.97  ohms, 
and  the  total  current  is  28°  25'  ahead  of  the  impressed 
pressure.  The  total  current  flowing  when  the  impressed 
pressure  is  100  volts  is  9.12  amperes,  and  it  requires 
109.7  volts  to  cause  10  amperes  to  flow.  When  10  am- 
peres are  flowing,  the  pressure  at  the  terminals  of  the 
branched  circuit  is  47.4  volts,  and  the  currents  which  flow 


O    2.5    Cl 


Solution   &i 
Fig.  81 

through  the  branches  are  5.9  and  4.3  amperes  respec- 
tively. The  pressure  across  the  first  part  of  the  circuit  is 
66  volts.  To  maintain  a  pressure  of  100  volts  on  the 
branched  part  of  the  circuit  requires  an  impressed  press- 
ure of  231.5  volts.  With  this  pressure,  21.1  amperes 
flow  through  the  circuit. 

The  second  method  of  solution  for  parallel  circuits 
may  be  applied  to  circuits  like  those  included  in  the 
above  example.  Figure  81  shows  the  solution  for  ex- 


206         ALTERNATING  CURRENTS. 

-. 

ample  bl  made  by  that  method.  In  this  it  is  assumed 
that  100  volts  is  impressed  upon  the  branched  part  of  the 
circuit.  Then  lay  off  a  length  OX  on  the  horizontal  axis 
representing  100  volts  and  mark  OC^  —  ^§-  =  2.5,  which 
is  the  current  in  the  first  branch.  The  current  in  the 
second  branch  is  90°  in  advance  of  the  pressure,  and  is 
represented  by  OC^  which  is  vertical  and  2nrfstlE(=  5) 
units  in  length.  The  resultant  of  these  currents  is  OC, 
which  is  5.6  units  in  length.  The  impressed  pressure 
measured  across  the  terminals  of  the  entire  circuit  is 
the  resultant  of  the  100  volts  at  the  terminals  of  the 
branched  part  of  the  circuit,  and  ttie  pressure  required 
to  pass  5.6  amperes  through  the  inductance  Ll  =  .01 
henry.  The  line  representing  the  latter  pressure  is 
perpendicular  to  the  line  representing  the  current  in 
the  circuit.  Drawing  a  semicircle  on  OX,  and  from  the 
intersection  of  OC  with  the  semicircle  drawing  a  line  to 
X  gives  the  direction  of  this  pressure.  The  magnitude 
of  the  pressure  is  27r/Z1(7=28  volts.  This  pressure 
must  be  laid  off  from  X  to  E,  and  the  total  impressed 
pressure  is  represented*  by  OE.  This  shows  that  when 
100  volts  is  maintained  at  the  terminals  of  the  branched 
circuit,  76  volts  must  be  impressed  on  the  total  circuit. 
The  resistances  of  the  circuit  may  be  calculated  from 
the  data  thus  found,  as  also  can  the  pressure  required  to 
maintain  a  certain  current  through  the  circuit. 

Figure  82  shows  the  solution  of  example  c  by  this 
method.  As  before,  the  pressure  at  the  terminals  of 
the  divided  circuit  is  assumed  to  be  100  volts  for  the 
purposes  of  the  solution.  This  is  laid  down  as  OX,  and 
a  semicircle  is  drawn  upon  the  line  as  a  diameter. 


METHODS    OF    SOLVING   PROBLEMS. 


207 


Tan  $3  is  equal  to  zero,  so  that  the  current  in  the 
first  branch  is  laid  off  on  OX  to  B,  a  distance  of  12.5 
units.  Tan  $2=  .45,  as  shown  by  calculation,  and  the 
line  OA'  is  laid  off  at  that  angle  from  OX.  From  O 
on  this  line,  OBJ  is  laid  off  equal  to  the  current  in  the 

second    branch,  or  -  OX.      The  resultant  of  the  lines 

OB  and  OB'  is  OB",  which  represents  the  total  cur- 
rent in  the  circuit.  OA"  is  the  equivalent  active  press- 


12.5 


=  -3825 


O  C  =  105.5 


D 

Fig.  82 


100 


ure  in  the  circuit.  The  total  pressure  impressed  on  the 
circuit  is  the  resultant  of  the  pressure  impressed  on 
the  divided  circuit,  the  active  pressure  due  to  resistance 
Rv  and  the  reactive  pressure  due  to  L±  and  sv  The 
active  pressure  required  to  pass  current  OB"  through 
R1  is  represented  by  OC,  which  is  equal  to  CR^.  The 
reactive  pressure  is  perpendicular  to  this  and  is  equal  to 


208         ALTERNATING  CURRENTS. 

2  TrfL-iC —  ;    it    is  represented   by    the  line  CD 

27r/y1 

The  pressure  impressed  on  the  parallel  circuit  is  repre- 
sented by  the  line  DE,  which  is  equal  and  parallel  to 
OX.  The  closing  line,  OE,  represents  the  impressed 
pressure  E  on  the  circuit  when  the  current  is  C,  and 

the  impedance  of  the  circuit  is  —  =  10.97.     The  angle 

of  lag  is  the  angle  COE  =  —  28°  25'. 

56.  An  Analytical  Method. — The  problems  just  solved 
graphically  may  also  be  readily  solved  analytically.* 

In  Sect.  49  it  has  been  shown  that  current,  press- 
ure, and  impedance  may  be  determined  in  magnitude 
and  relative  phase  by  means  of  a  polar  diagram.  Thus, 
in  Fig.  83,  suppose  OX  to  be  the  initial  line  and 
OA',  OA" ,  and  OA'"  to  be  pressures  or  impedances  in 
series,  or  currents  or  conductances  (reciprocals  of  im- 
pedances) in  parallel,  which  are  represented  in  relative 
phase  by  the  angular  positions,  and  in  magnitude  by 
the  lengths  of  the  lines.  It  has  just  been  shown  that 
the  resultant  of  two  or  more  similar  electrical  quantities 
may  be  found  by  treating  their  representative  lines  as 
vectors ;  such  vectors  may  be  combined  by  algebrai- 
cally adding  the  vertical  and  horizontal  components  of 
the  individual  lines,  by  which  means  the  vertical  and 
horizontal  components  of  the  resultant  are  determined. 
If  a',  b'\  a",  b"  •  and  a1",  b'"  are  the  horizontal  and 


*Steinmetz  on  Complex  Quantities,  Proceedings  of  the  International 
Congress  held  at  Chicago  in  1893,  p.  33 ;  Steinmetz  on  Hysteresis,  Trans. 
Amer.  Insi.  E.  E.,  Vol.  n,  p.  576;  Tail's  Quaternions,  Hardy's  Quater- 
nions, etc. 


METHODS    OF    SOLVING   PROBLEMS. 


209 


vertical  components  of  OA',  OA",  and  OA"f,  and  A,  B, 
the  components  of  the  resultant,  then 

A  +  B  =  (a1  +  a"  +  a'")  +  (b'  +  b"  +  #"), 
or         A+B=(a!  +  b')  +  (a"  +  W)  +  (*'"  +  £'"). 

In  this  expression  there  is  nothing  to  distinguish  the 
horizontal  from  the  vertical  components ;  or,  in  general, 

Y 


a"  a'" 


a' 
Fig.  83 


there  is  nothing  to  indicate  the  angular  positions  of  the 
components,  or  of  the  lines  represented  by  them,  with 
reference  to  the  initial  line.  To  fully  indicate  the  mag- 
nitude and  position  of  a  line  by  its  rectangular  com- 
ponents, we  must  abandon  the  methods  of  algebra  for 
geometric  processes.  Therefore  we  may  consider,  for 


210 


ALTERNATING    CURRENTS. 


the  moment,  that  the  components  /  and  u  of  the  vector 
A  (Fig.  83  a)  both  lie  on  the  initial  line  OX,  but  in  order 


Fig.  83  a 

that  /  and  u  may  determine  the  vector  A,  u  must  be 
rotated  90°.  To  indicate  such  a  rotation,  a  prefix  such 
as  i  may  be  used.  Then  A  will  be  represented  in  mag- 
nitude and  angular  position  by  the  expression 

/  +  iu, 

where  the  sole  function  of  the  letter  i  is  to  indicate 
that  the  component  u  stands  90°  from  the  initial  line, 
and  the  addition  is  geometric.  Inasmuch  as  iu  is  posi- 
tive, u  has  been  rotated  ahead  90°;  —iu  would  indicate 
that  u  had  been  rotated  in  a  negative  direction  90°,  or  u 
is  measured  downwards  (the  negative  direction)  from 
the  origin.  If  t  and  in  are  both  negative,  they  are  both 
measured  in  the  negative  direction  ;  hence,  if  /  +  iu  be 
multiplied  by  —  i,  there  results  —  t  —  iu,  t  and  u  are 
both  reversed  in  direction,  and  the  vector  line  OA  is 
rotated  180°  (Fig.  84);  it  —  u  means  that  the  line  has 


METHODS   OF   SOLVING   PROBLEMS. 


211 


been  rotated  forward  90°,  since  t  is  positive  but  stands 
at  90°  from  the  initial  line,  and  u  is  negative;  —  it  +  u 
means  that  the  line  has  been  rotated  back  90°.  Finally, 
multiplying  by  i  means  advancing  the  vector  line  90°, 


A  (-90) 


A  (180°) 


Fig.  84 

for  i(t  4-  hi)  =  it  +  (  +  i)(iu),  and  as  i2  indicates  rotation 
twice  forward,  i2u  becomes  —  u,  and  therefore  i  (t  +  in) 
is  geometrically  equal  to  it  —  u ;  also  multiplying  by 
—  i  means  turning  the  vector  back  90°,  for  —  i(t  +  iu) 


212  ALTERNATING   CURRENTS: 

=  —  it  4-  (—  *)(«/),  and  as  —  z2  indicates  rotation  for- 
ward 90°  and  back  90°,  —  z2//  =  u,  and  there  results 
-  it  4-  «. 

The  vector  expressing  a  sine  wave  may  now  be  repre- 
sented in  magnitude,  as  heretofore  shown,  by 


in  phase  by 

in  phase  and  magnitude  by  the  complex  quantity 

A  (cos  <f>  +  i  sin<£), 

and  also,  as  just  indicated,  by  the  equivalent  quantity 

t+iu. 

The  addition  of  the  vectors  given  in  the  first  illustration 
now  becomes 

A+iB=  (a1  +  a"  +  a'")  +  i(b'  +  b"  +  6"'). 

Suppose  it  is  desired  to  combine  two  impedances 
which  are  in  series,  as  /'  and  I",  Fig.  85,  in  which  /,  /' 
and  r"y  I"  are  the  rectangular  components  (resistances 
and  reactances).  A  sinusoidal  pressure  acting  on  I' 
must  evidently  overcome  a  self-inductive  reactance 
equal  and  opposite  to  Ol1  and  a  resistance  Or\  while 
the  pressure  acting  on  I"  must  overcome  a  capacity 
reactance  equal  and  opposite  to  Ol"  and  a  resistance 
Or".  Then,  if  R,  H,  are  the  components  of  the  result- 
ant (I'"),  this  equation  may  be  written 

I"'=R  +  *//=  (/  +  r")  +  i(l'  -  I"}, 


METHODS   OF   SOLVING   PROBLEMS. 


213 


from  which  the  magnitude  and  phase  position  of  the  re- 
sultant impedance  may  be  found. 

If  the  impedances  are  in  parallel,  their  reciprocals 
must  be  combined,  in  which  case  the  resultant  is  the 
reciprocal  of  the  impedance  (conductivity)  of  the  divided 
circuit.  It  is  evident  that  the  components  of  the  con- 
ductivity (reciprocal  of  impedance)  will  not  be  equal  to 


Fig.  85 

the  reciprocals  of  the  components  of  the  impedance. 
The  components  of  the  conductivity  must  therefore  be 
found  in  terms  of  the  impedance  components.  If  p,  X 
are  the  conductivity  components  and  r,  I  the  impedance 
components  (resistance  and  reactance)  of  a  single  cir- 
cuit, we  may  write  by  the  principles  of  geometric  mul- 
tiplication, 

K=l-  =  ^i\  =  -1—  (a) 

I  r±il 

The  numerical  values  of  the  first  and  second  terms  of 
the  right-hand  member  of  the  expression  K  —  p  T  i  X  are 


214  ALTERNATING   CURRENTS. 

respectively  proportional  to  the  active  current  and  watt- 
less current  in  a  circuit.  When  a  circuit  contains  induc- 
tive reactance  only,  /  is  essentially  positive,  but  the 
wattless  current  lags  90°  behind  the  active  current,  so 
that  X  is  essentially  negative.  When  a  circuit  contains 
capacity  reactance  only,  /  is  essentially  negative,  but  the 
wattless  current  leads  the  active  current  by  90°,  so  that 
X  is  essentially  positive.  When  a  circuit  contains  both 
inductive  and  capacity  reactance,  the  signs  of  /  and  X 
are  dependent  upon  the  relative  magnitudes  of  the 
inductance  and  capacity.  The  value  of  K  may  be  writ- 
ten in  a  general  manner 


1       r+  ili  —  ils 
To  reduce  the  equation 


to  a  more  convenient  form,  the  numerator  and  denomi- 
nator of  the  right-hand  member  may  be  multiplied  by 

r  T  H,  whence 

..v  _          r^il          _  r  T  il 
~  (r  T  il)  (r  ±  il)  ~  r^  +  I*' 

since  z'2  indicates  the  operation  which  is  equivalent  to 
multiplying  by  —  I. 

*s  7 

Hence,  p  T  i\=  - 


but  the  impedance  (/)  is 


Therefore, 


METHODS  OF  SOLVING  PROBLEMS.         215 

or  p  =  ji  and  X  =  — 

Since  r,  /,  and  /  are  known  or  can  be  determined  from 
the  conditions  presented,  problems  relating  to  parallel 
circuits  can  now  be  solved. 

Returning  to  the  example,  if  p',  X',  and  p",  X"  are 
the  components  of  the  conductivities  of  two  parallel 
circuits  having  impedances  I'  and  /"  (Fig.  85), 

j          it 
P>-*'  =  JTt-'jrt 

and  p»  +  ,V' 

and  if  p,  X  are  the  components  of  the  final  conductivity, 


The  intrinsic  sign  of  i\  depends  upon  the  relative  signs 

/'  I" 

and  magnitudes  of  —  and  —^     The  impedance  of  the 

circuit  will  be  the  reciprocal  of  the  conductivity  thus 
found. 

When  7  is  the  impedance  and  K  the  conductivity  of 
a  circuit,  we  write  I=r±il  and  K  =  p  T  i\  according  to 
the  principles  of  geometric  addition  which  assert  that 
the  sum  of  two  sides  of  a  triangle  taken  in  consecutive 
directions  is  equal  to  the  third  side.  This  is  entirely 
opposed  to  the  ordinary  conceptions  of  algebra  or 
arithmetic. 

These  processes  which  enable  us  to  find  the  joint  im- 
pedance of  parallel  or  series  circuits  when  the  elements 
of  the  individual  parts  of  the  circuits  are  known,  equally 


216 


ALTERNATING   CURRENTS. 


enable  us  to  find  the  impedance  of  any  combination  of 
such  circuits  by  computations  which  are  almost  as 
simple  and  rapid  as  those  which  would  be  used  in  deal- 
ing with  a  continuous-current  system.  Also,  when  the 
impedances  of  any  combination  of  circuits  have  been 
obtained,  it  is  possible  to  find  the  pressures  in  any  por- 
tion when  a  sinusoidal  current  is  flowing  and  to  find  the 
current  when  a  sinusoidal  pressure  is  applied. 

The  meaning  of  the  terms  in  the  expression  for 
impedance  and  conductivity  may  be  explained  by: 
multiplying  (r±il)  by  C  (current  in  the  circuit),  when 
it  is  evident  that  rC  is  the  active  pressure  and  1C  the 
component  of  pressure  acting  against  the  reactance ; 
and  also  by  multiplying  (p  T  fa)  by  Et  (pressure  im- 
pressed on  the  circuit),  when  pEt  is  the  active  com- 
ponent of  the  current  and  \Et  the  wattless  current. 

The  following  is  a  recapitulation  of  the  formulas  for 
the  analytical  solution,  by  geometric  processes,  of  prob- 
lems relating  to  alternating-current  circuits. 


GEOMETRIC  EQUATIONS. 


=  /,;  (cos  0z  +  i  sin  0Z). 


ALGEBRAIC  EQUATIONS. 


/»  f 

=  ,/*  =  £,  ,=  x/=  =  A 

/_ 

cos  0      sin  0 

cos  0       sin  0 
tan  0  =  -  =  -• 


C=^=EK. 


METHODS    OF   SOLVING   PROBLEMS. 


For  convenience  in  computations  the  geometric  equa- 
tions should  be  set  out,  for  example,  as  follows : 


r 

IL 

4 

I'   =    r' 

+  if 

I"  =    r' 

+    UL" 

-    if,  if 

Jill  rlll 

-    i!"1 

Ix    =Sr 

+  *S/£ 

-*s/. 

=  rx  ±  i/*. 

57.  Illustration  of  Analytical  Method. — In  illustration 
of  this  method,  solutions  to  some  of  the  problems  to  be 
found  on  the  foregoing  pages  are  given.  7,  K,  and  <f> 
are  used  to  represent  respectively  impedance,  conduc- 
tivity, and  lag. 

Series  Circuits  (see  Sect.  50). 

Forming  the  equations  for  the  non-inductive  and  in- 
ductive coils  in  problems  c,  h,  i,  and  j  of  Sect.  50. 


A  =  S  +  »8 
/ii  =    5  +  *  o 

/    =  10  +  i  8 
c=ioo  =  ioo         g 
/       12.8      ' 


tan  0  =  —  =  .8,  0  =  38°  40'. 


/=. 


=  12.8. 


cos  0      sin  0 

,£=:  10 /=   10  X    12.8  =    128. 


As  the  prefix  of  the  reactance  term  in  the  expression 
for  /  is  4-  it  the  angle  (j)  is  positive. 


A.       /i    =0+  112 

/n  =  o  —  z  10 

/   =  O  +  i  2 
100 

C=  — =  50. 


/I    =0  +  JI2.5 

/n  =0-  £12.5 

/      =0+0 


tan  0  =  -  =  oo,  0  = 


sin  0       i 
E  —  10  x  2  =  20. 

7=o. 


ALTERNATING  CURRENTS. 


tan  0  =  — -,  0  =  —  24°  14' 
io 


r 


=   10.96. 


j.   /i     =  io  +  io 
/ii  =    o  +  *  8 

/111  =     0-JI2-5 

7~^To-i  4.5  cos0    sm<^ 

.£  =  io  X  10.96  =  109.6. 
10.96 

Here  the  prefix  of  the  reactance  term  in  the  expres- 
sion for  /is  —  i,  hence  the  angle  <£  is  negative. 

Parallel  Circuits  (see  Sect.  52). 
Applying  the  formula  for  conductance, 


10 


100  +  O 

=  o  —  i 


64  +  0 


K    —  .  i  —  i  .125 


tan  0=  - — 5  —  1.25;   0  =  51°  20'. 


cos  0       sin  0 
/=J.:=cos0  =  6> 

A-         P 
E  —  io  x  6.25  =  62.5. 


0  is  positive  in  this  case,  as  —  i  in  the  expression  for 
K  shows  that  the  current  lags  behind  the  pressure. 


io 


_ -°954  _ 


e.  A  i  —  —  t 
100+  16 

ioo  +  16 

tan  0  — 

—   .7072. 
•1349 

K  -    8 

10 

0  = 

35°  '6'. 

64  +  ioo 

64  +  ioo 

K  = 

•I349-.i65. 

K  =  .1349  -  z.  0954 

COS0 

I- 

-i-  -  6.06. 

C  =  —  =  ioo  K 

=  16.5. 

.165 

1 

E  = 

io  x  6.05  =  60.6. 

f-  K     I0 

o 

tan0  = 

.0488 

100  +  0 

IOO  +  O 

.I390-'3509' 

K      I0    ! 

12.5 

0  = 

-  19°  20'. 

ioo  +  156 

ioo+  156 

K- 

.1390  _  l 

K  =.1390  +  2.04 

88 

COS0 

I- 

—  =  6.8. 

C=  —  =  ioo  K 

=  H-7- 

.147 

I 

E  — 

io  x  6.8  =  68. 

METHODS   OF   SOLVING   PROBLEMS. 


219 


<£  is  negative  in  this  case,  as  +  i  in  the  expression  for 
K  shows  that  the  current  leads  the  pressure. 

4 


K\\  — 


-  +  i 
loo  +  156        100  -f-  156 

K    =.1610  —  2.0487 
C=-^=iooJS-=  16.8. 


A-i   =- 


10 


8.33 


100  +  69.4 

IO 


-f  i 


100  +  69.4 
8-33 


loo  +  69.4        loo  +  69.4 
K    =  .1181  -  *o 

<7=-^°-  =  100  K  —  1 1. 8. 


tan0  =     "J     =  .1O2C,  0  =  16°  so'. 
.1610 

^=il^=.i68. 


/=  —  =  5.95. 

/- o          J    ^w 

£=iox  5.95  =  59.5. 


tan0  = 


.1181 


,0=0. 


A-=^=.n8i. 


COS0 


/  = 


=  8.47- 


.1181 

1  =  10  x  8.47  =  84.7. 
The  effect  of  the  reactances  in  this  circuit  is  note- 
worthy. 


TT           o             ,•" 

•045 

64  +  0 

64+0 
1    i     I2'5 

tan  0  —  —  oo. 
o 

0  =  90°. 

.04  5 
K—       °  —  .041;. 

*'"-  156  +  0 

156  +  o 

K"  —  o  —  i.OAH 

sin  0 

/=  22.2. 
E  =  10  X  22.2  =  222. 

Series  and  Parallel  Circuits  Combined  (see  Sect.  55). 

tan  <f>AB  =  -^,    <f>AS  =  35°  1 6'. 


a.  KA   —  .0862  —  i  .0345 

KB    =  .0487  —  1.0609 


=  -I349-  z-0954 


IAB  =  6.06  (cos0AJ5  +  z'sin  0A^ 
—  6.06  (.8165  +  i  .5774) 
=  4-95  +  *'3-5°- 

JAB  =4-95  +  *'3-5° 
IG   —  10  +  i  8 

/     =14.95  +  111.50 


COS0 


=  6-°6' 


-- 

.165 


/=  1  8.  8. 


=  10  x  18.8=  188. 


22O 


—  .025  —  i  o 
=  -o      +  i  .0503 


ALTERNATING  CURRENTS. 

tan  (f)AB  =  '—-—,  (f>AB  =  -  63°  32'. 


-  -025  +  i  .0503 


/AB  =  17.8  (cos  (f>AB  +  z  sin 
=  17.8  (.4456 -i.  8960) 

SAB  =  7-937  ~  ^5- 749 
/cr    =  o         +  j    5.027 

•^     —  7-937  ~  *  10.922 


COS0 


=  -V  =  i7.8. 

.0561 


7-937 


0^-540  o'. 


.fi=io  x  13.5  = 


THE   MAGNETIC   CIRCUIT   OF   ALTERNATORS.     221 


CHAPTER   V. 

THE    MAGNETIC    CIRCUIT    OF    ALTERNATORS. 

58.  Losses  in  an  Alternator.  —  The  principles  which 
enter  into  the  design  of  alternators  have  already  been 
thoroughly  set  forth  in  Vol.  I.  and  in  the  first  chapter 
of  these  notes.  There  are,  however,  certain  peculiar 
features  in  the  magnetic  circuits  and  the  methods  of 
applying  the  windings  to  alternators  that  require  con- 
siderable modifications  of  the  deductions  found  in 
Vol.  I.  These  will  now  receive  examination  in  detail. 
As  in  continuous-current  dynamos  (Vol.  I.,  p.  248),  the 
internal  losses  of  alternators  are  caused  by : 

1.  C^R  loss  in  the  conductors  on  armature  and  field. 

2.  Foucault  or  eddy  currents  in  armature  cores  and 
field. 

3.  Foucault  or  eddy  currents  in  armature  conductors. 

4.  Hysteresis  in  armature  cores. 

5.  Friction  of  bearings  and  brushes,  and  air  friction. 

In  well-designed  continuous-current  dynamos,  the 
pole  pieces  usually  cover  not  less  than  two-thirds  of  the 
armature  surface  (Vol.  I.,  pp.  167  and  272).  In  alterna- 
tors, the  poles  usually  cover  about  one-half  of  the  arma- 
ture surface,  or  even  less  (Sect.  5).  This  would  make 
it  appear,  upon  a  superficial  examination,  that  the  field 


222  '        ALTERNATING  CURRENTS. 

ampere-turns,  and  therefore  the  field  losses,  must  be 
much  greater  in  the  alternator.  However,  since  alterna- 
tor armatures  are  made  proportionally  larger  in  diameter, 
in  order  to  give  space  for  the  windings  and  to  avoid 
excessive  magnetic  leakage,  the  proportional  excita- 
tion really  required  need  not  be  much  increased  when 
the  magnetic  circuit  is  well  designed.  In  the  same 
way,  while  not  much  more  than  one-half  of  the  arma- 
ture surface  is  covered  with  wire,  the  surface  for  wind- 
ing is  made  much  larger  by  increasing  the  diameter, 
while  the  number  of  revolutions  is  not  much  reduced. 
Consequently,  the  pressure  produced  in  a  given  length 
of  conductor  is  entirely  commensurable  in  the  two 
classes  of  machines.  This  is  illustrated  by  the  table  on 
page  14,  and  by  the  following  machines  of  three  excel- 
lent makers : 

1.  Two-pole  continuous-current  dynamo  of  60  K.W. 
output;  armature  core,  15"  diameter,  15"  long;  winding 
requires  185  Ibs.  wire;   C^Ra  loss,  2.4  percent;  speed, 
900  revolutions  per  minute;  fields  with  15,000  ampere- 
turns  at  full  load  ;  field  wire,  470  Ibs.  ;  total  C2Rf  loss, 
1.7  per  cent ;  total  weight  of  the  machine,  10,000  Ibs. 

2.  Four-pole  continuous-current  dynamo  of  75  K.W. 
output;   armature  core;  22"  diameter,  17^'  long;  wind- 
ing requires   235   Ibs.   wire;  C2l?a  loss,    3.0  per  cent; 
fields  with  15,000  ampere-turns  at  full  load;  field  wind- 
ing requires  756  Ibs.  wire ;  total  C^Rf  loss,  2.3  per  cent; 
total  weight  of  machine,  n,ooo  Ibs. 

3.  Alternating-current  dynamo  of  70  K.W.  output ; 
armature  core,  24"  diameter,  i8-|-"  long;  winding  requires 
70  Ibs.  wire  ;  C2Ra  loss,  1.6  per  cent ;  speed,  1050  revolu- 


THE    MAGNETIC   CIRCUIT   OF   ALTERNATORS.     223 

tions  per  minute  ;  fields  with  25,000  ampere-turns  at  full 
load  ;  field  winding  requires  725  Ibs.  wire  ;  total  C*Rf 
loss,  2.4  per  cent ;  total  weight  of  the  machine,  9500  Ibs. 

The  table  on  pages  224  and  225  gives  data  of  two 
English  alternators  built  by  the  firm  of  Elwell  &  Par- 
ker,* and  of  five  American  machines  of  about  equal 
capacity.  All  but  one  of  these  have  drum  armatures, 
but  in  the  English  machines  they  are  stationary  and  sur- 
round the  revolving  poles,  while  in  the  American  ma- 
chines the  poles  surround  the  revolving  armature.  The 
armature  of  one  of  the  American  machines  is  of  the  disc 
type.  All  of  the  American  machines  were  built  about 
1890,  but  all  except  one  have  since  been  superseded  by 
a  more  substantial  construction. 

59.  Copper  Losses.  —  We  may  safely  say  that  the  per- 
centage C2R  losses  given  upon  pages  108  and  138  of 
Vol.  I.  can  be  taken  as  a  limit  towards  which  practice 
in  the  design  of  alternators  is  tending.  The  fact  that 
the  copper  is  divided  among  many  cores  increases  the 
length  of  wire  on  alternator  fields  for  a  given  magnet- 
izing power  as  compared  with  continuous-current  fields, 
and  the  C^R  losses  allowed  are  usually  somewhat  greater 
than  the  tabular  values  and  sometimes  reach  more  than 
twice  those  values.  (Examples  :  Kapp  30  K.W.  alterna- 
tor with  5.0  per  cent  loss  in  the  field  windings  and  2.8 
per  cent  in  the  armature  conductors  ;  Ferranti  112  K.W. 
alternator  with  2.75  per  cent  loss  in  the  field  windings ; 
and  General  Electric  300  K.W.  alternator  with  2.0  per 
cent  loss  in  the  field  windings.)  On  the  other  hand, 
the  losses  may  be  brought  by  careful  designing  to 

*  See  Thompson's  Dynamo- Electric  Machinery,  4th  ed.,  p.  668. 


224 


ALTERNATING   CURRENTS. 


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THE   MAGNETIC   CIRCUIT   OF   ALTERNATORS.      225 


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226 


ALTERNATING   CURRENTS. 


approach  the  average  values  given  in  the  tables.  (Ex- 
amples :  Stanley  40  K.W.  alternator  with  equal  losses 
of  1.2  per  cent  in  fields  and  armature,  and  180  K.W. 
alternator  with  .7  per  cent  loss  in  fields  and  2.0  per 
cent  loss  in  armature;  Mordey  75  K.W.  alternator 
with  1.5  per  cent  loss  in  fields  and  2.3  per  cent  in 
armature;  Hopkinson  30  K.W.  alternator  with  1.6  per 
cent  loss  in  armature  and  2.9  per  cent  in  fields.)  These 
examples  may  be  extended  to  an  indefinite  number,  but 
those  given  illustrate  the  conditions.  The  following 
table  may  therefore  be  taken  as  showing  the  percentage 
C2R  losses,  which  represent  the  best  present  practice, 
and  towards  which  all  good  practice  in  the  design  of 
alternators  must  eventually  tend.  At  the  present  time 
it  is  not  unusual  to  build  alternators  of  greater  capacity 
than  75  K.W.  with  upwards  of  double  the  field  loss  given 
in  this  table,  notwithstanding  the  fact  that  the  table 
gives  values  for  the  loss  in  the  fields  of  alternators 
which  are  considerably  greater  than  the  tabular  values 
for  continuous-current  machines  (Vol.  L,  p.  138). 


KILOWATTS. 

PER  CENT  IN  ARMATURE. 

PER  CENT  IN  FIELDS. 

30 

35 

2.4 
2-3 

2.8 
2.6 

40 

2.25 

2.4 

50 
60 

2.2 
2.15 

2.2 
2.O 

75 

2.1 

1.8 

oo 

2.O 

1.7 

150 

200 

i.g 

1.6 
1.6 

Table  based  on  cold  resistances  at  about  25°  C.  (75°  F.). 


THE   MAGNETIC   CIRCUIT   OF   ALTERNATORS.     22/ 

60.  Foucault  Current  Losses.  —  Foucault  currents  in 
alternator  cores  must  cause  much  greater  loss  than  in 
ordinary  continuous-current  machines.  Thus,  the  arma- 
tures of  two-pole  continuous-current  machines  of  con- 
siderable capacity  seldom  exceed  1500  revolutions  per 
minute,  which  makes  twenty-five  complete  magnetic 
cycles  in  the  core  per  second.  In  alternators,  the  com- 
mercial frequency  commonly  lies  between  40  and  140, 
while  the  greater  number  of  machines  are  built  for  fre- 
quencies between  60  and  135.  The  number  of  magnetic 
cycles  of  the  armature  core  is  evidently  equal  to  the  fre- 
quency. Since  the  heating  in  the  core  discs  which  is 
caused  by  foucault  currents  is  proportional  to  the  square 
of  the  number  of  cycles  per  second,  it  becomes  particu- 
larly important  that  the  discs  composing  alternator  arma- 
tures be  well  insulated  from  each  other.  It  is  therefore 
poor  policy  to  depend  upon  oxide  alone  for  insulation, 
and  tissue  paper  should  always  be  placed  between  the 
discs,  and  a  cardboard  be  inserted  at  intervals.  All 
burrs  caused  by  punching  the  discs  or  truing  the 
surface  of  the  core  should  be  carefully  avoided  or 
removed. 

Foucault  currents  in  the  pole  pieces  are  felt  quite 
severely  in  some  types  of  alternators,  but  the  loss  caused 
by  them  can  always  be  brought  within  reasonable  limits, 
in  well-designed  machines,  by  making  the  pole  faces  of 
laminated  iron  which  is  cast  into  the  frame.  (Compare 
Vol.  L,  p.  154.)  Machines  having  smooth  armature 
cores  and  surface  windings  do  not  ordinarily  require  this 
precaution,  but  in  machines  having  toothed  armature 
cores,  laminated  poles  are  quite  important. 


228  ALTERNATING   CURRENTS. 

The  loss  due  to  foucault  currents  in  armature  conduct- 
ors of  a  fixed  size  is  much  greater  for  alternating  than 
for  continuous-current  dynamos,  on  account  of  the  more 
frequent  and  sudden  changes  in  the  strength  of  the 
field  through  which  they  pass.  However,  since  alternat- 
ors are,  in  general,  built  for  considerably  higher  press- 
ures than  continuous-current  machines  designed  for  a 
similar  duty,  the  conductors  on  the  alternator  armatures 
are  of  proportionally  smaller  cross-section.  This  reduces 
the  eddy  currents  to  such  an  extent  that  they  are  not 
particularly  noticeable  except  in  very  large  or  special 
machines.  The  practice  of  winding  armature  coils  with 
copper  ribbon  set  on  edge  (the  broad  side  parallel  with 
the  lines  of  force)  also  tends  to  decrease  eddy  currents. 
In  very  large  machines  built  to  generate  a  pressure  not 
exceeding  2000  volts,  armature  conductors  of  a  large 
cross-section  become  essential,  but  the  uniform  practice 
of  building  large  alternator  armatures  with  imbedded 
conductors  avoids  all  difficulty  from  eddy  currents. 
(Compare  Vol.  I.,  p.  153.) 

61.  Hysteresis  Losses.  —  The  effect  of  hysteresis  in 
iron  core  armatures  is  proportional  to  the  number  of 
magnetic  cycles  per  second,  and  is  therefore  much 
greater  in  alternator  armatures,  for  a  given  magnetic 
density,  than  in  those  of  continuous-current  machines. 
There  are  only  two  ways  of  decreasing  the  hysteresis 
loss  per  cycle  and  per  unit  volume  :  (i)  by  reducing  the 
induction  ;  (2)  by  improving  the  quality  of  the  iron  used 
in  the  core.  Reducing  the  induction  serves  to  decrease 
the  foucault  current  loss  also,  and  is  therefore  doubly  ad- 
vantageous. As  a  consequence,  the  induction  in  alterna- 


THE   MAGNETIC   CIRCUIT   OF   ALTERNATORS.     229 

tor  armatures  is  sometimes  made  as  low  as  4000  lines  of 
force  per  square  centimeter,  and  seldom  exceeds  7000 
lines  per  square  centimeter.  A  fair  average  value  is 
about  5000  lines  per  square  centimeter.  This  is  just 
about  one-half  the  average  value  for  continuous-current 
armatures  (Vol.  I.,  p.  1 1 1).  Halving  the  induction  tends 
to  quarter  the  foucault-current  loss  for  an  equal  number 
of  cycles  per  second.  But,  as  already  shown,  the  num- 
ber of  cycles  in  alternator  armatures  commonly  varies 
from  three  to  six  times  the  maximum  number  occurring 
in  continuous-current  machines  of  the  same  capacity,  so 
that  the  special  precautions  for  insulating  the  discs  can- 
not be  neglected.  This  is  rendered  more  necessary  by 
the  fact  that  reducing  the  induction  requires  the  arma- 
ture to  be  increased  in  size.  In  the  same  -way,  the 
hysteresis  loss  varies  nearly  in  the  proportion  of  B  to 
the  1.6  power,  or  in  other  words,  halving  the  induction 
decreases  the  hysteresis  loss  per  unit  volume  to  one- 
third.  On  the  other  hand,  since  the  cross-section  of 
iron  must  be  increased  to  decrease  the  induction  per 
square  centimeter,  it  is  not  possible  to  bring  the  hyste- 
resis loss  to  the  limit  attainable  in  continuous-current 
machines.  This  makes  the  careful  selection  and  hand- 
ling of  the  iron  designed  to  enter  alternator  cores  of 
special  importance.  Some  manufacturers  of  electrical 
machinery  not  only  select  their  armature  iron  with  great 
care  on  this  account,  but  anneal  all  armature  discs  after 
they  have  been  assembled  and  the  surface  of  the  core  has 
been  turned  up.  For  this  purpose,  the  cores  are  carefully 
taken  down  after  all  machine  work  on  the  discs  has  been 
completed  and  the  discs  are  annealed,  after  which  the 


230  ALTERNATING   CURRENTS. 

cores  are  again  assembled.  This  method  of  handling 
the  iron  removes  the  hardening  effect  of  the  tooling  con- 
sequent upon  turning  up  the  cores  (Vol.  I.,  pp.  52  and  73). 
62.  Armature  Ventilation.  —  Since  the  hysteresis  and 
foucault-current  losses  are  greater  in  alternator  arma- 
tures, it  is  necessary  that  more  opportunity  be  given  for 
cooling  than  is  the  case  in  continuous-current  machines. 
The  real  effect  of  the  velocity  of  rotation  upon  cooling 
has  never  been  thoroughly  determined,  but  the  experi- 
ments of  Rechniewski  *  seem  to  show  a  cooling  effect 
that  is  roughly  proportional  to  the  velocity.  Conse- 
quently, on  account  of  their  high  surface  velocity,  alter- 
nator armatures  are  more  efficiently  cooled  than  are 
continuous-current  machines.  Internal  ventilation  is 
also  rendered  more  effective  by  reason  of  the  high  sur- 
face velocity.  For  the  purpose  of  internal  ventilation 
a  revolving  armature  is  virtually  converted  into  a  cen- 
trifugal blower,  which  sucks  in  air  at  its  centre,  along 
the  shaft,  and  ejects  it  from  the  surface.  For  this  pur- 
pose air  ducts  are  made,  which  run  through  the  core 
near  the  shaft,  and  which  communicate  with  the  surface 
through  radial  ducts.  The  rotation  of  the  armature 
then  causes  a  continuous  circulation  of  air  through  the 
ducts,  which  is  proportional  in  volume  to  the  surface 
velocity  of  the  armature. f  Since  the  conductors  cover 
only  a  portion  of  the  surface  and  do  not  cross  the  ends 
of  drum  armatures,  the  cores  may  be  easily  arranged  for 
internal  ventilation,  and  advantage  is  usually  taken  of 

*  Bulletin  de  la  Societe  Internationale  des  Electriciens,  1892;  Electrical 
World,  Vol.  19,  p.  336;   also  Textbook,  Vol.  I,  p.  109. 
t  Kent's  Mechanical  Engineer's  Pocket  Book,  p.  516. 


THE   MAGNETIC   CIRCUIT  -OF   ALTERNATORS.     231 

this.  In  the  case  of  revolving  disc  armatures,  wings 
may  be  so  placed  as  to  blow  the  air  across  the  face 
of  the  disc.  Ring  armatures  and  pole  armatures  are 
more  difficult  to  arrange  for  effective  internal  ventila- 
tion, and  therefore  they  are  frequently  left  unventilated. 
When  the  fields  revolve,  the  poles  cause  a  vigorous 
circulation  of  air,  which  serves  in  lieu  of  the  blower 
action  of  a  revolving  armature. 

63.  Armature  Radiating  Surface.  —  With  all  precau- 
tions to  avoid  excessive  heating  in  alternator  cores  they 
still  tend  to  heat  to  a  higher  temperature  than  the 
cores  of  continuous-current  armatures.  It  then  becomes 
a  matter  of  some  concern  to  determine  the  possible 
amount  of  energy  which  may  be  expended  in  the  arma- 
ture conductors  without  placing  an  excessive  additional 
burden  upon  the  cooling  surface.  There  is  no  valid  rea- 
son for  admitting  a  higher  temperature  limit  in  alter- 
nator armatures  than  is  allowed  in  continuous-current 
machines  (Vol.  I.,  p.  105).  As  this  fact  becomes  en- 
tirely admitted  by  all  manufacturers  of  alternating-cur- 
rent machinery,  the  large  C2R  losses  in  alternator 
armatures  that  are  now  common  will  rapidly  disappear. 
On  account  of  the  large  amount  of  heat  caused  by  core 
losses  which  must  be  radiated,  it  is  not  safe  to  allow  an 
armature  C2R  loss  of  one  watt  for  each  square  inch  of 
cooling  surface.  In  common  practice,  for  each  watt  of 
C2Ra  loss  an  average  of  from  i|  to  2  square  inches 
of  outside  winding  surface  is  allowed.  This  constant  is 
sometimes  made  much  larger,  but  seldom  smaller  except 
in  disc  armatures.  In  the  latter  the  core  losses  need 
not  be  provided  for,  the  whole  capacity  of  the  radi- 


232  ALTERNATING    CURRENTS. 

ating  surface  may  be  utilized  to  carry  off  the  C2Ra  loss, 
and  it  is  possible  to  decrease  the  radiation  surface  to 
considerably  less  than  one  square  inch  per  watt. 

64.  Density  of  Current  in  Armature   Conductors.  — 
The  density  of  current  in  conductors  wound  upon  iron- 
core   armatures   in    good   practice    should   usually  not 
much  exceed  one  ampere  to  from  500  to  600  circular 
mils.     (Compare  continuous-current    dynamos,  Vol.  I., 
p.  1 10.)     Many  iron-core  armatures  are  wound  with  wire 
having  not  much  more  than  one-half  this  cross-section 
per  ampere,  but  their  makers  allow  an  excessive  rise  of 
temperature  in  the  machines  during  working.     In  disc 
armatures  without  iron,  the  constant  may  be  greatly  re- 
duced without  danger.    Thus  the  Mordey  75  K.W.  alter- 
nators have  a  current  density  in  the  armatures  of  one 
ampere  to  about  380  circular  mils,  and  the  current  density 
in  Ferranti  alternator  armatures  is  about  one  ampere  to 
335  circular  mils.     Even  with  such  a  great  current  den- 
sity, these  machines  are  comparatively  cool  in  operation. 

65.  Field  Radiating  Surface.  —  Since  the  field  cores 
of  alternators   are  usually  quite  thin,  the  windings  are 
often  of  a  depth  equal  to  one-half  the  thickness  of  the 
cores.     At   the   same  time  this   depth  is  generally  no 
greater  than  that  on  many  continuous-current  dynamo 
fields.     The  same  radiating  constant  can  therefore  be 
safely  used,  that    is,  from  .35    to  .40  watt   per  square 
inch  of  the  outer  surface  of  the  winding  (Vol.  L,  p.  142). 
Since  the  depth  of  the  winding  generally  bears  a  large 
ratio  to  the  thickness  of  the  core,  the  energy  radiated 
per  square  inch  of  core  surface  is  much  greater.     The 
ratio  of  winding  depth  to  thickness  of  core  is  also  widely 


THE   MAGNETIC   CIRCUIT   OF   ALTERNATORS.     233 

variable,  and  the  radiation  constant  must  be  based  upon 
the  external  surface  of  the  windings.  The  field  cores 
should  therefore  be  made  of  such  a  length  that  the  area 
of  the  external  surface  of  the  windings  in  square  inches 
may  be  from  two  to  three  times  the  C*Rf  loss  in  watts. 
The  form  of  the  fields,  and  therefore  the  effect  of  the 
iron  in  conducting  away  the  heat  in  the  field  windings, 
has  a  considerable  effect  on  the  allowable  value  of  the 
constant,  as  is  also  the  case  in  continuous-current  ma- 
chines. The  area  of  the  field  cores  is  given  by  the  for- 
mula, A  —  Na— ,  where  Na  is  the  useful  armature  flux  re- 
Bt 

quired  from  one  pole,  v  is  the  leakage  coefficient,  and 
Bf  the  induction  desired  in  the  field  core.  The  pole 
width  is  determined  by  the  mechanical  and  electrical 
conditions  which  fix  the  pitch,  and  only  the  length  of 
the  pole  faces  may  be  altered  to  vary  Bt.  If  the  winding 
depth  exceeds  two  inches,  it  is  likely  to  cause  injurious 
heating  in  the  lower  layers. 

66.  Leakage  Coefficient. — The  value  of  the  leakage 
coefficient  is  quite  large  in  most  types  of  alternators- 
It  probably  averages  1.5  in  alternators  with  surface 
wound  drum  or  ring  armatures  and  with  poles  set  in  a 
circle  either  without  or  within  the  armature.  In  pole  ar- 
matures it  is  doubtless  equally  as  great,  but  in  machines 
having  well-designed,  toothed,  ring  or  drum  armatures, 
and  small  air  spaces,  the  leakage  coefficient  may  be  less. 
In  machines  with  ring  or  disc  armatures  in  which  poles 
of  opposite  sign  are  ranged  alternately  on  either  side 
of  the  armature,  the  value  of  the  coefficient  may  be  as 
great  as  2.0.  In  machines  of  the  Mordey  type,  where 


234  ALTERNATING   CURRENTS. 

the  poles  on  each  side  of  a  disc  armature  are  of  the 
same  sign,  or  of  the  Stanley  so-called  inductor  type,  the 
leakage  coefficient  becomes  unity;  that  is,  practically 
all  of  the  lines  of  force  set  up  in  the  magnetic  circuit 
cut  the  armature  conductors,  and  are  therefore  useful 
in  developing  electric  pressure. 

The  calculation  of  the  leakage  coefficient  of  alterna- 
tors may  be  carried  out  upon  the  same  methods  as  those 
used  for  continuous-current  machines.* 

67.  Determination  of  the  Number  of  Armature  Con- 
ductors.— The  formula  E  =  —^ — ~£  (Sect.  5)  may  be 

io8  x  60 

put  into  the  form  E— ^,  since  ~  is  equal  to  the 

io8  60 

frequency,  f.  Taking  a  proper  value  for  K,  as  already 
explained,  gives  the  effective  value  of  E.  In  a  well- 
designed  machine  of  the  usual  American  types,  the 
value  of  K  is  about  i.i,  which  is  the  value  for  a 
machine  which  gives  a  sinusoidal  electric  pressure  and 
in  which  the  differential  action  is  negligible.  The  con- 
ditions of  service  usually  fix  the  values  of  E  and  /  in 
any  particular  case,  and  the  equation  then  contains  two 
dependent  variables,  N  and  5.  The  ratio  of  these  is 
determined  from  the  form  and  dimensions  of  the  arma- 
ture which  it  is  desired  to  make.  The  number  of  poles 
is  limited  by  constructive  considerations  and  the  impor- 
tance of  keeping  the  leakage  within  reasonable  limits. 
On  the  latter  account  the  poles  must  not  come  too  near 
together.  With  this  in  view  the  frequency  cannot  be 
increased  beyond  a  certain  limit  by  increasing  the  num- 

*  Textbook,  Vol.  I,  p.  133;  also  Electrical  Engineer,  Vol.  18,  p.  163 
et  seq. 


THE   MAGNETIC   CIRCUIT   OF   ALTERNATORS.     235 

ber  of  poles,  without  carrying  the  periphery  velocity  of 
the  armature  beyond  the  safe  limit.  Thus,  suppose 
a  machine  be  designed  with  a  ten-pole  stationary  field, 
and  its  drum  armature  be  designed  to  drive  at  the 
safe  limit  of  velocity.  If  it  be  desired  to  increase  the 
frequency  by  20  per  cent,  two  poles  must  be  added  to 
the  field  and  the  armature  revolutions  kept  constant,  or 
the  armature  must  be  speeded  up  20  per  cent.  The 
latter  is  not  permissible  on  account  of  mechanical 
safety.  If  the  poles  are  already  as  close  together  as 
economy  admits,  in  order  to  increase  the  number  of 
poles  the  pitch  circle  must  be  increased  in  diameter, 
which  requires  a  proportional  increase  of  the  armature 
diameter.  This  again  calls  for  an  increase  of  the  sur- 
face velocity,  the  revolutions  per  minute  remaining 
constant.  Hence  in  any  type  of  machine  a  limit  of 
frequency  may  be  reached  which  cannot  be  safely 
exceeded.  The  limiting  frequencies  in  the  ordinary 
types  of  machines  designed  for  commercial  service  are 
from  100  to  150.  In  the  Morel ey  type  the  limit  is  much 
higher,  since  the  poles  may  be  very  close  together  and 
cause  no  leakage,  and  structural  considerations,  only, 
set  the  limit.  Commercial  frequencies  are  all  less  than 
150  and  many  are  less  than  100,  so  that  all  practical 
types  of  alternators  may  be  used  in  commercial  service. 
Fixing  the  frequency  of  a  machine  and  the  periphery 
velocity  and  revolutions  of  the  revolving  part,  fixes  both 
the  diameter  of  the  armature  and  the  number  of  poles.' 
The  diameter  of  the  armature  fixes  the  space  for  arma- 
ture coils.  With  the  winding  space  fixed,  the  value  of 
5"  is  determined  from  the  number  of  insulated  conduct- 


236  ALTERNATING   CURRENTS. 

ors  of  requisite  area  which  can  be  properly  placed  on 
the  armature.  It  is  quite  usual  to  wind  alternator  arma- 
tures with  only  two  layers  of  wire,  but  this  must  be  de- 
termined in  any  particular  case  by  many  conditions 
affecting  the  design.  The  value  of  5  being  deter- 
mined, the  length  of  the  armature  must  be  made  such 
that  the  necessary  total  magnetization  may  be  set  up 
in  the  field  cores  and  armature  without  forcing  the 
magnetic  density  to  too  high  a  value.  Finally  the  ratio 
of  radiating  surface  to  C2Ra  loss  should  be  checked. 

It  is  well  to  consider  here  the  effect  upon  the  out- 
put of  an  alternator,  of  making  a  change  in  the  number 
of  armature  conductors.  The  electric  pressure  devel- 
oped in  the  coils  due  to  their  cutting  lines  of  force  is 
proportional  to  the  number  of  turns  in  the  coils.  The 
electric  pressure  of  self-induction,  on  the  other  hand, 
is  proportional  to  the  square  of  the  number  of  turns. 
Hence,  increasing  the  number  of  armature  turns  beyond 
a  certain  limit  may  actually  decrease  the  output  of  the 
machine.  It  is  desirable  to  determine  the  number  of 
turns  that  will  give  maximum  output.  If  Ll  represents 
the  effective  or  working  self-inductance  for  each  arma- 
ture conductor,  then  the  total  effective  self-inductance 
of  the  armature  is  L  =  S^L^  In  the  same  way,  if 
EI  represents  the  electric  pressure  developed  per  con- 
ductor, the  total  pressure  developed  in  the  armature 
is  E  =  vS-Zf.  From  the  fundamental  formula 


we  get 


THE   MAGNETIC   CIRCUIT   OF   ALTERNATORS.     237 


or 

This  may  be  put  into  the  form 


CR  =  VS2^2  -  47r2f2C2S*L2, 

if  the  external  circuit  to  which  the  alternator  is  con 
nected  is  non-inductive.  The  term,  CR,  is  the  active 
pressure  in  the  alternator  circuit,  and  it  is  desirable 
for  this  to  be  a  maximum  for  a  given  armature.  Dif- 
ferentiating the  equation  with  respect  to  5  and  solving 
for  a  maximum,  we  get  the  following  : 

d(CR)         S(E2  -  8  Tr^C^L*) 

= 


hence  E?  -  8  >jr2f2C2L2S2  =  o.     Or  CR  is  a  maximum 
when  J-? 

s= 


In  this  it  is  assumed  that  the  armature  resistance  is 
small  compared  with  that  of  the  external  circuit,  which 
is  always  the  case  in  efficient  working.  For  E1  may  be 
substituted  its  value 


and  the  expression  for  vS  at  maximum  output  becomes 

N 


where  A  is  a  constant  depending  upon  the  type  and 
dimensions  of  the  machine.  If  K  has  a  value  of  i.i,  the 
value  of  A  is  practically  25  x  io~10.  The  final  form  of 
the  variable  portion  of  the  expression  giving  the  maxi- 
mum economical  value  of  6"  is  striking.  Its  numerator 


238  ALTERNATING   CURRENTS. 

is  the  total  useful  magnetization  due  to  the  fields,  which 
passes  through  an  armature  coil,  and  its  denominator 
is  the  magnetization  passing  through  the  coil  due  to  the 
current  in  each  of  its  own  conductors.*  This  criterion 
shows  that  the  armatures  of  some  old-style  magneto 
alternators  had  too  much  wire  for  economy;  that  is, 
they  would  have  supplied  a  larger  pressure  to  a  non- 
inductive  circuit  if  fewer  conductors  had  been  placed 
on  their  armatures. 

When  the  external  circuit  is  inductive,  as  is  usually 
the  case,  the  number  of  armature  turns  which  gives  a 
maximum  active  pressure  is  smaller  than  when  the  ex- 
ternal circuit  is  non-inductive.  If  Sf  be  the  number  of 
conductors  giving  the  maximum  active  pressure  when 
the  external  circuit  has  self-inductance  Le,  and  6"  be  the 
number  of  conductors  giving  the  maximum  pressure 
when  the  external  circuit  is  non-inductive,  then 

L     or          =I 


r 

In  the  latter  expression  S2L1  is  the  self-inductance  of 
the  armature  when  wound  with  the  proper  number  of 
turns  to  give  a  maximum  active  pressure  when  the 
external  circuit  is  non-inductive.  When  Le  is  greater 

than  S*L-i,  the  right-hand  member  of  the  expression  for 

SI 

-—  becomes  imaginary.     It  is  impossible  to  put  so  many 

turns  on  commercial  alternator  armatures  as  the  criterion 
shows  would  give  the  greatest  output,  since  the  ques- 


Picou,  Machines  Dynamo- Alectriques,  p.  271. 


THE   MAGNETIC   CIRCUIT   OF   ALTERNATORS.     239 

tion  of  regulation  in  constant-pressure  alternators  de- 
mands that  the  fall  of  pressure  in  the  armature  due  to 
resistance  and  inductance  shall  be  as  small  as  possible.* 
68.  Example  of  Armature  Calculation.  —  Suppose  it 
is  desired  to  design  a  50  K.W.  alternator  of  the 
American  type,  for  1000  volts  terminal  pressure  and 
50  amperes,  at  a  frequency  of  60,  using  a  smooth  core 
armature  with  surface  windings.  Assume  K  to  be  i.i  ; 
then,  adding  10  per  cent  to  the  terminal  pressure  to 
allow  for  loss  of  pressure  in  the  armature  due  to 
loss  and  inductance,  we  have 


i  833,000,000. 

2.2  X  60 

We  may  take  1000  revolutions  as  a  satisfactory  maxi- 
mum speed  at  which  to  aim.  One  thousand  revolu- 
tions and  a  frequency  of  60  gives  a  fractional  number 
of  poles  while  the  number  must  be  an  even  integer. 
Taking  900  revolutions  gives  exactly  eight  poles, 
which  is  satisfactory.  Then  taking  the  safe  periphery 
velocity  as  6000  feet  per  minute  makes  the  diameter 
of  the  armature  a  trifle  over  two  feet.  Call  the  diame- 
ter of  the  core  two  feet.  The  periphery  of  this  is 
75.4  inches.  Somewhat  more  than  one-half  of  this 
winding  space  should  be  occupied  by  wire,  say  46 
inches.  Each  coil  must  generate  one-eighth  of  the 
pressure,  or  I37-|-  volts,  if  the  armature  is  connected 
in  series.  The  diameter  of  the  pitch  circle  for  the 
poles  may  be  set  approximately  as  25  inches,  and  its 

*  Compare  Kapp's  Dynamos,  Alternators,  and  Transformers,  p.  377; 
Steinmetz,  Trans.  Amer.  hist.  E.  E.,  Vol.  12,  p.  326. 


240         ALTERNATING  CURRENTS. 

circumference  is  then  78.5  inches.  The  combined 
width  of  the  poles  should  be  somewhat  less  than  half 
of  this,  or  say  38  inches.  This  makes  each  pole  4^ 
inches  or  12.1  centimeters  in  width.  Since  46  inches 
of  the  armature  circumference  are  occupied  by  wire, 
about  27.4  inches  are  left  for  the  openings  in  the 
centres  of  the  coils  when  •£•  inch  is  allowed  for  insu- 
lation between  the  coils.  This  makes  the  openings 
approximately  3^-  inches  wide.  The  cross-section  of 
the  armature  conductors,  allowing  525  circular  mils  to 
the  ampere,  must  be  26,250  circular  mils.  This  is  the 
cross-section  of  a  No.  6  B.  &  S.  Ga.  wire  which  has  a 
diameter  of  178  mils  when  double  cotton-covered.  Two 
hundred  and  fifty-eight  of  these  will  go  into  46  inches 
in  one  layer,  but  the  number  of  armature  conductors 
must  be  a  multiple  of  twice  the  number  of  coils,  or 
16.  Two  hundred  and  fifty-six  is  the  multiple  which 
is  nearest  to  258.  This  makes  32  conductors  or  16 
turns  per  coil,  and  gives  N  the  value  of  3,254,000 
lines  of  force.  Allowing  an  average  induction  of  5500 
under  the  pole  faces  makes  the  length  of  the  pole  faces, 
practically,  19^  inches.  This  is  too  great  a  length  to 
be  practical  in  a  machine  of  the  capacity  under  consid- 
eration, and  two  layers  of  wire  must  be  put  on  the 
armature,  thus  reducing  N\  or  by  rolling  the  wire  into  a 
rectangular  form  and  placing  it  on  the  armature  surface 
on  its  edge,  it  may  be  made  to  occupy  less  surface  and 
more  conductors  may  be  put  on  the  armature ;  in 
which  case  either  the  air  gap  induction  or  the  dimen- 
sions of  the  armature  may  be  reduced,  or  these  may  be 
reduced  together.  It  is  therefore  quite  common  to  wind 


THE   MAGNETIC   CIRCUIT   OF   ALTERNATORS.     241 

alternator  armatures  with  special  rectangular  wire  or 
ribbon,  and  we  will  take  a  ribbon  which  is  250  x  80  mils 
in  cross-section,  which  gives  an  area  equivalent  to  about 
2 5, 500  circular  mils.  When  this  is  triple  cotton-covered, 
its  dimensions  may  be  taken  to  be  270  x  100  mils,  and 
464  of  these  will  wind  into  a  space  46.4  inches  wide. 
This  makes  58  conductors  or  29  turns  per  coil.  This 
gives  Nthe  value  of,  practically,  1,795,000  lines  of  force, 
and  makes  the  length  of  the  pole  faces  approximately 
12  inches  when  the  average  induction  in  the  air  gap 
is  5000.  It  remains  then  to  fix  the  exact  dimensions  of 
the  armature  and  pole  pieces.  Taking  the  diameter 
of  the  armature  core  as  24  inches,  and  adding  double 
the  thickness  of  insulation  gives,  say,  24.25  inches. 
This  makes  allowance  for  two  layers  of  japanned  canvas 
and  a  layer  of  mica  under  the  coils.  The  winding 
diameter  is  24.25  inches,  and  the  circumference  is 
76.18  inches.  From  this  is  subtracted  46.40  inches, 
and  2  inches,  which  is  the  space  occupied  by  the  con- 
ductors and  the  insulation  between  the  coils,  and  there 
remains  27.78  inches  for  the  spaces  within  the  coils. 
The  coils,  made  so  as  to  turn  down  at  the  ends  of  the 
core,  therefore  have  the  following  approximate  dimen- 
sions (also  Fig.  86) :  outside  length,  A  =  ig^  inches ; 
inside  length,  ^=14  inches;  outside  width,  C=g^ 
inches ;  inside  width,  D  =  3!  inches.  This  leaves  \ 
inch  between  the  coils  which  may  be  filled  by  a  strip 
of  vulcanized  fibre  or  paraffined  wood.  The  total 
length  of  wire  is  approximately  950  feet,  which  has 
an  approximate  weight,  insulated,  of  80  pounds  and  a 
cold  resistance  of  .40  ohm.  The  C2Ra  loss  is  therefore 
R 


242 


ALTERNATING    CURRENTS 


1000  watts,  or  2.0  per  cent,  based  on  the  cold  resist- 
ance, which  is  not  far  from  the  tabular  value  (page  226); 
the  total  winding  surface  is  76.2  x  19.8=  1509  square 
inches,  and  this  gives  more  than  i|-  square  inches  per 
watt  C2Ra  loss,  which  is  satisfactory.  Since  the  arma- 
ture conductors  number  464,  it  is  required  that  N  be 
equal  to  1,795,000  lines  of  force.  One-half  this  num- 


Fig-.  86 

ber  of  lines  passes  through  each  magnetic  circuit  in  the 
armature.  Putting  Ba  as  4000  makes  the  cross-section 
of  the  armature  core  about  225  square  centimeters,  or 
35  square  inches.  The  length  of  iron  in  the  core  may 
be  assumed  to  be  12  x  .80,  or  9.6  inches.  The  depth 
of  the  core  discs  must  therefore  be  about  3|  inches, 
or  the  inner  diameter  of  the  discs  is  i6|-  inches. 
The  external  finished  diameter  of  the  armature  is 


THE    MAGNETIC   CIRCUIT   OF   ALTERNATORS.     243 

24.25  +  .540  +  .164  =  24.95.  This  allows  31  mils  for 
the  thickness  of  insulation  under  the  bands,  and  51  mils 
for  the  wire  in  the  bands.  Wire  of  5  1  mils  diameter,  or 
1  6  B.  &  S.  gauge,  is  used  on  account  of  the  high  pe-  * 
riphery  velocity  of  the  armature.  Allowing  a  little 
under  ^  inch  (210  mils)  for  mechanical  clearance  makes 
the  diameter  of  the  polar  circle  25.37  inches,  or  25! 
inches.  The  circumference  of  the  polar  circle  is  there- 
fore 79.7  inches.  The  pitch  of  the  poles  is  9.95  inches, 
and  the  distance  between  their  tips  is  5.2  inches. 

69.  Armature  Self-Inductance.  —  The  self  -inductance 
of  a  smooth-core  alternator  armature  may  be  approxi- 
mately estimated  from  the  magnetic  and  electric  data  of 
the  machine.  The  reluctance  of  each  magnetic  circuit 
must  be  calculated  exactly  as  in  the  case  of  a  continu- 
ous-current multipolar  dynamo,  in  order  to  determine 
the  field  windings.  In  the  American  type  of  alternator, 
the  reluctance  to  be  overcome  by  the  magnetic  press- 
ure of  each  field  core  belongs  to  that  part  of  the  circuit 
which  lies  between  the  lines  AA'  and  BBf  in  Fig.  87. 
Calling  that  reluctance  P,  the  ampere-turns  for  each 


_ 

field   core  are  in   number  nc  =  --      The   reluctance 

1.25 

in  the  different  parts  of  the  magnetic  circuit  met  by 
lines  of  force  which  are  set  up  by  the  armature  turns 
when  the  fields  are  excited,  may  be  assumed  to  be  equal 
to  the  reluctance  in  the  same  parts  of  the  circuit  met 
by  the  lines  set  up  by  the  field  coils.  The  number  of 
lines  of  force  set  up  in  the  portion  of  the  magnetic  cir- 
cuit between  the  lines  AA'  and  BB1  by  a  unit  current 
in  one  armature  coil,  the  centre  of  which  is  directly 


244  ALTERNATING   CURRENTS. 

T   o  C    Q* 

under  a  pole  face,  is  -i— ; •? — J,  where  6\  is  the  number  of 

conductors  in  the  coil,  and  is  equal  to  twice  the  number 
of  turns  in  the  coil.  Each  one  of  these  lines  of  force 
in  completing  its  circuit  must  link  another  armature 
coil,  so  that  we  may  say  that  the  number  of  lines  of 


Fig-.  87 

force  set  up  by  each  pair  of  coils  is  — - — j~-     The  self- 
induction  of  the  pair  of  coils  is  therefore 

1.25  sf 

J^l  =  p; g> 

2  X  P  X   IO8 

since  S1  is  equal  to  the  number  of  turns  in  two  coils 
The  self-induction  of  the  whole  armature  is  equal  to  Lt 
multiplied  by  the  number  of  pairs  of  coils,  when  the 
armature  is  connected  in  series,  or 

L  =    1.25  Sfp  : 

2  X  P  X  IO8 


THE   MAGNETIC   CIRCUIT   OF   ALTERNATORS.     245 

When  the  armature  is  connected  with  the  halves  in 
parallel,  the  inductance  is  one-fourth  as  great  as  is 
given  by  this  formula ;  but  in  the  case  of  two  similar 
armatures  built  for  the  same  pressure  and  output,  the 
one  connected  with  the  halves  in  parallel  has  twice  as 
many  conductors  in  each  coil  as  has  the  other  armature, 
and  their  self-inductances  are  equal.  If  the  value  of  P, 
in  the  preceding  example,  is  taken  as  .004,  the  self- 
inductance  is  shown  by  substitution  to  be  L  =  .021 
henry. 

The  effect  of  the  ampere-turns  of  the  armature  coil, 
within  the  ordinary  load  limits  of  a  smooth-core  arma- 
ture, will  not  greatly  alter  the  permeability  of  the  highly 
magnetized  fields,  so  that  L  may  be  taken  to  be  approx- 
imately constant  with  varying  loads,  provided  the  arma- 
ture pressure  be  kept  constant. 

The  path  of  the  lines  of  force  set  up  by  the  armature 
coils  has  been  assumed  to  be  the  same  as  the  path  of 
those  set  up  by  the  field  magnets.  This  is  approxi- 
mately true  for  machines  with  smooth  drum  armature 
cores,  or  with  coreless  armatures,  and  the  real  effect  of 
the  armature  turns  upon  the  number  of  lines  of  force 
in  the  magnetic  circuits,  is  to  increase  or  decrease  the 
number  that  would  exist  were  the  armature  turns  absent, 
rather  than  to  set  up  an  independent  magnetization. 
The  effects  of  armature  reactions  and  of  self-induction 
are  therefore  closely  related.  In  the  case  of  machines 
with  toothed  armature  cores,  the  reluctance  in  the  path 
of  the  magnetization  due  to  the  field  is  materially 
smaller  than  when  the  cores  are  smooth,  and  hence  it  is 
to  be  expected  that  the  self-inductance  of  toothed-core 


246 


ALTERNATING   CURRENTS. 


armatures  will  be  large.  If  the  teeth  are  T-shaped  as 
in  Fig.  88,  the  reluctance  measured  around  the  path 
of  the  lines  of  force  set  up  by  the  armature  coils  may 
be  materially  smaller  than  the  reluctance  measured 
along  the  path  of  the  magnetization  due  to  the  field 
coils.  This  is  due  to  the  effect  of  the  leakage  from 
tooth  to  tooth.  Consequently,  the  self-inductance  of  an 
armature  having  T-shaped  teeth  which  are  close 
together  may  be  expected  to  be  very  large.  In  some 


Fig-.  88 

such  machines  which  are  arranged  to  have  a  specially 
large  armature  self-inductance  in  order  to  obtain  a  self- 
regulating  constant-current  machine,  as  in  the  Stanley 
arc  light  alternator,  the  inductance  may  be  as  much 
as  two  or  three  henrys. 

70,  Armature  Reactions.  —  The  armature  reactions  of 
alternators  by  no  means  cause  as  serious  consequences 
as  those  of  continuous-current  machines.  When  the 
current  of  the  armature  is  in  exact  phase  with  the  im- 
pressed pressure,  the  armature  current  has  compara- 


THE   MAGNETIC   CIRCUIT   OF   ALTERNATORS.     247 

tively  little  opportunity  to  affect  the  field  magnetism. 
When  the  armature  conductors  are  directly  between  the 
pole  pieces,  the  instantaneous  current  is  zero,  and  there- 
fore at  this  point  the  armature  has  no  effect  upon  the 
field  magnetism.  When  the  coils  have  moved  through 
one-half  the  pitch,  a  sheet  of  current  at  its  maximum 
value  flows  directly  under  the  pole  faces.  This  current 
has  such  a  direction  that  its  magnetic  effect  tends  to 
crowd  the  lines  of  force  of  the  field  into  the  trailing  tips 


MAGNETIZATION. 
C.  RESULTANT  FIELD. 


Fig.  89 

of  the  poles  (Fig.  89).  Hence  the  field  is  weakened  on 
account  of  the  increased  reluctance  of  the  magnetic 
circuit.  This  effect  is  probably  not  very  marked  in  the 
usual  forms  of  alternators,  since  the  reluctance  of  the 
path  occupied  by  the  armature  or  cross-lines  of  force  is 
quite  large.  The  distortion  and  consequent  weakening 
of  the  field  may  be  reduced  by  cutting  a  slot  longitudi'- 
nally  across  the  pole  faces,  or  by  some  of  the  methods 
described  in  Vol.  I,  Chap.  VI. 


248 


ALTERNATING    CURRENTS. 


When  the  armature  current  is  out  of  phase  with  the 
impressed  electric  pressure,  the  conditions  are  quite 
different.  Suppose  that  the  phase  of  the  current  is 
retarded  on  account  of  self-induction.  When  the  cen- 
tres of  the  coils  are  under  the  poles,  the  current  is  not 
zero,  but  has  an  instantaneous  value  which  depends  upon 
the  amount  of  retardation.  This  current  in  a  generator 
is  in  such  a  direction  that  its  magnetic  effect  opposes 
that  of  the  field  (Fig.  90).  As  the  coils  move,  this 
opposing  effect  merges  into  the  cross-effect  already 
indicated.  When  the  machine  under  consideration  is 
operating  as  a  motor,  the  current  under  the  poles  evi- 


Fig.   90 

dently    tends    to    strengthen    the    fields    instead    of   to 
weaken  them. 

If  the  current  is  in  advance  of  the  phase  of  the 
impressed  pressure,  the  armature  of  a  generator  tends 
to  strengthen  the  field  magnetism  when  the  coils  are 
directly  under  the  poles  (Fig.  91).  A  motor  armature 
under  like  conditions  tends  to  weaken  the  fields.  This 
tendency  of  the  armature  current,  when  in  advance 


THE    MAGNETIC   CIRCUIT   OF   ALTERNATORS.     249 

of  the  impressed  pressure,  to  strengthen  the  fields,  may 
be  taken  advantage  of  to  make  an  alternator  completely 
self-regulating,  or  even  self-exciting,  through  the  action 
of  its  armature  current.  This,  however,  requires  the 


Fig.  91 

use  of  a  condenser  attached  across  the  armature  termi- 
nals to  give  the  proper  lead  to  the  current,  which  is 
undesirable.  The  opposing  and  cross-magnetic  effects 
of  the  retarded  armature  currents  of  alternating  gen- 
erators, when  operating  under  usual  conditions,  cause 
the  external  characteristic  to  slope  toward  the  hori- 
zontal axis.  This  effect  must  be  added  to  the  slope 
of  the  characteristic  caused  by  true  and  inductive 
resistance  in  the  armature.  It  is  often  difficult  to  dis- 
tinguish between  the  effects  of  armature  reactions 
proper  and  of  self-induction,  and  they  are  sometimes 
treated  as  alike.* 

The   quantitative   effect   of  the   current,  and   of  the 
angle    of    lag,    on    armature    reactions    is    not    readily 

*  Kapp's    Dynamos,    Alternators,    and    Transformers,    p.    394;    and 
Hawkins  and  Wallis'  The  Dynamo. 


250  ALTERNATING   CURRENTS. 

determined.  It  is  evident  that  the  effect  is  a  peri- 
odic one  which  depends  for  its  relative  instantaneous 
values  upon  the  instantaneous  positions  of  the  coils 
with  reference  to  the  poles  ;  and  which  depends  fur- 
ther for  its  actual  instantaneous  and  average  values 
on  the  current  strength,  the  angle  of  lag,  and  the 
shape  of  the  current  curve.  Doubtless  the  relative 
shapes  of  armature  coils  and  pole  pieces  also  enter 
the  relation.  Since  the  effect  of  the  reactions  is  peri- 
odic, it  is  difficult  to  determine  its  exact  result  in  any 
particular  case,  by  any  means  except  that  of  experi- 
ment. The  field  frames  are  fairly  large  masses  of  iron, 
and  they  do  not  respond  rapidly  to  changes  in  their 
magnetic  surroundings.  This  inertia  is  caused  by  the 
effect  of  foucault  currents  and  the  considerable  induct- 
ance of  the  field  windings,  which  tend  to  suppress 
sudden  magnetic  changes.  It  is  therefore  safe  to 
assume  in  general  that  the  discernible  effect  of  arma- 
ture reactions  is  an  average  of  the  instantaneous  values. 
The  instantaneous  value  of  the  back  turns  of  each  coil 
at  any  moment  is  A/2  nC  sin  (a—  <£)cos  a,  where  n  is  the 
number  of  turns  of  each  coil,  C  is  the  effective  value 
of  the  current  which  is  assumed  to  be  sinusoidal,  and 
<f>  is  the  angle  of  lag.  This  expression  may  be  averaged 
between  the  limits  a  ==  o°  and  a  =  TT,  with  the  result 
that  the  back  turns  appear  to  be  approximately  equal  to 
2.22  nCsin  </>.*  This  formula  purports  to  give  the  num- 
ber of  ampere-turns  to  be  added  to  each  pole  on  account 
of  back  turns,  and  the  result  is  positive  or  negative  as 
the  current  lags  or  leads  ;  but  it  does  not  include  the 

*  Compare  Kapp's  Dynamos,  Alternators,  and  Transformers,  p.  412. 


THE   MAGNETIC   CIRCUIT    OF   ALTERNATORS.     251 

effect  of  cross-magnetization,  which  is  sometimes  con- 
siderable but  is  difficult  to  predetermine.  The  method 
of  figuring  the  effect  of  inductance  has  already  been  in- 
dicated (Sect.  69),  and  all  the  corrections  necessary  can 
nowbe  made  in  computing  field  windings.  This  is  car- 
ried out  as  explained  in  Vol.  I.  (p.  143  et  scq.},  due 
attention  being  given  to  modifying  conditions  already 
explained. 

71.    Field  Excitation  of  Alternators.  —  The  windings 
of  the  field  magnets  of  alternators  are  usually  classified 


Fig.  92 

according  to  their  arrangement  in  circuit.  The  prin- 
cipal divisions  are  three  :  separately  excited,  self -excited, 
compositely  excited;  so-called,  respectively,  when  the 
magnetizing  current  is  supplied  from  an  external  source 
(Fig.  92),  when  it  is  supplied  through  a  rectifying  com- 


252          ALTERNATING  CURRENTS. 

mutator  from  the  armature  of  the  machine  under  con- 
sideration (Fig.  93),  or  when  these  two  arrangements 
are  combined  (Fig.  94).*  Self-excited  alternators  may 
again  be  divided  into  series-wound  and  shunt-wound, 
depending  upon,  first,  whether  the  whole  current  is 
rectified  and  led  through  a  comparatively  small  num- 
ber of  turns  around  the  field  magnets  (Fig.  95),  or, 
second,  whether  only  a  portion  of  the  current  is  recti- 
fied and  led  through  a  shunt  circuit  many  times  around 
the  magnets  (Fig.  96).  (Example :  Zipernowsky  alter- 
nator.) Of  these  divisions  of  self-excited  alternators, 
the  shunt-wound  is  the  more  common.  In  this,  either 
the  whole  pressure  of  the  armature,  or  that  of  one  or 
more  coils,  may  be  impressed  directly  upon  the  rectify- 
ing commutator,  by  means  of  a  transformer  attached 
to  the  armature  (Fig.  97).  (Examples  :  Westinghouse 
and  Zipernowsky  alternators.)  Figure  97  illustrates  the 
arrangement  when  the  fields  rotate. 

Evidently  a  third  division  might  be  added  to  these, 
which  would  be  a  combination  of  the  other  two,  or  a 
compound  winding  in  which  both  the  shunt  and  series 
field  currents  are  supplied  by  rectification.  This,  how- 
ever, would  require  two  rectifying  commutators,  which 
at  the  best  are  unsatisfactory,  and  for  other  reasons 
would  not  prove  practical.  To  gain  the  result  for  which 
compounding  is  used  in  continuous-current  dynamos, 
the  composite  winding  is  used.  That  is,  the  alternator 
is  externally  excited  to  its  normal  pressure  on  open  cir- 
cuit and  the  internal  losses  are  compensated  by  series 
ampere-turns  from  self-excitation.  The  self-exciting 

*  Compare  Text-book,  Vol.  I.,  p.  136. 


THE   MAGNETIC   CIRCUIT   OF   ALTERNATORS.     253 


Fig.  93 


Fig.  94 


254 


ALTERNATING   CURRENTS. 


Fig.  96 


THE    MAGNETIC   CIRCUIT   OF   ALTERNATORS.     255 

circuit  of  the  composite  winding  may  be  arranged  in 
various  ways.     Thus  the  armature  current  may  all  be 


Fig.  97  a 


Fig.  97  b 

rectified  for  use  in  excitation  (Fig.  98)  (example : 
Thomson-Houston  alternator),  or  the  armature  cur- 
rent may  pass  through  a  special  transformer  attached 


256  ALTERNATING   CURRENTS. 


THE   MAGNETIC   CIRCUIT   OF   ALTERNATORS.     257 

to  the  armature,  and  the  secondary  of  this  may  then 
supply  the  current  for  rectification  and  self-excitation 
(Fig.  99).  The  core  of  this  transformer  may  be  either 
independent  of  the  armature  core,  as  at  A  in  the  figure, 
or  may  consist  of  the  laminated  spider  or  other  portions 


Pig.  99 

of  the  armature  core.  (Example :  some  Westinghouse 
alternators.)  Again,  the  rectified  current  may  be  passed 
through  a  few  turns  of  wire  on  each  pole  (Fig.  100),  or 
all  the  necessary  series  turns  may  be  concentrated  upon 

s 


258 


ALTERNATING  CURRENTS. 


one  or  two  poles  (Figs.  98  and  101).  (Examples  :  West- 
inghouse,  Thomson-Houston,  and  General  Electric  alter- 
nators.) In  the  latter  case,  the  series  turns  must  always 
be  equally  divided  between  two  poles  with  symmetrical 
positions  when  the  armature  is  connected  with  its  halves 
in  parallel  (Fig.  102).  Composite  windings  may  be  ar- 
ranged with  the  self-excitation  in  a  shunt  circuit,  but  no 


Fig.  1OO 

advantage  is  gained  by  this  arrangement  over  complete 
self-excitation  in  shunt  or  by  separate  excitation.  This 
arrangement  is,  therefore,  not  used.  In  some  self-excit- 
ing alternators,  a  separate  set  of  exciting  coils  is 
wound  on  the  armature  and  connected  to  a  rectifying 
commutator.  These  may  be  wound  directly  with  the 


THE   MAGNETIC   CIRCUIT   OF   ALTERNATORS.     259 

main  armature  coils  or  across  a  chord  of  the  armature 
core  (Fig.  103).  (Example  :  old-style  Thomson-Houston 
alternator.)  The  compounding  may  be  effected  in  self- 
excited  alternators  by  means  of  shunt  and  series  trans- 
formers combined  as  in  Fig.  97  b,  which  shows  a  machine 
with  stationary  armature.  (Example  :  Ganz  alternators.) 


Fig.  1O1 

The  rectifying  commutator  in  every  case  has  as  many 
segments  as  there  are  poles  on  the  alternator,  and  alter- 
nate segments  are  connected  together,  making  two  sets 
(Fig.  104).  To  each  of  these  sets  one  of  the  alter- 
nating-current terminals  is  attached.  Brushes  bearing 
upon  the*  commutator  at  opposite  non-sparking  points 


ALTERNATING   CURRENTS. 


then  collect  a  rectified  current.  Various  devices  have 
been  employed  to  avoid  sparking  at  the  rectifying  com- 
mutator, but  in  American  machines  no  special  precau- 
tions are  taken.  (Examples  :  Westinghouse,  Thomson- 
Houston,  and  General  Electric  alternators.)  In  the 


Pig-.  1O2 

Zipernowsky  alternator,  built  by  Ganz  &  Co.  of  Buda- 
Pesth,  the  following  arrangement  of  the  commutator  is 
employed  :  Between  the  commutator  divisions  are  in- 
serted narrow  metallic  sectors  which  are  connected 
together.  Four  brushes  are  used,  two  on  eaph  side  of 


THE   MAGNETIC  CIRCUIT   OF   ALTERNATORS.    261 

the  commutator.  One  brush  of  each  pair  is  set  a  little 
in  the  lead  of  the  other,  and  the  pair  is  connected 
together  through  a  small  resistance.  The  leading 


Fig.  1O3 

brushes  are  connected  directly  to  the  circuits.  When 
the  commutating  point  is  reached  during  the  rotation 
of  the  commutator,  the  trailing  brushes  move  on  to 
intermediate  segments,  while  the  forward  brushes  are 


Fig.  1O4 


still  on  main  segments.  Hence  both  the  field  circuit 
and  supply  circuit  are  short-circuited  for  an  instant 
through  the  resistances  connecting  the  brushes  (Figs. 


262 


ALTERNATING   CURRENTS. 


105  and  106).  Short-circuiting  the  supply  circuit  has 
a  disadvantage,  but  if  a  transformer  is  used  for  excita- 
tion it  may  be  so  designed  that  no  harm  results.  The 
short-circuiting  of  the  field  circuit  is  claimed  to  give 
two  points  of  advantage  :  First,  it  allows  the  com- 
mutation to  be  effected  with  little  sparking;  second, 
upon  short-circuiting  the  fields,  their  self-induction 
tends  to  uphold  the  current  in  the  windings,  and  this, 


FIELD> 


r\j\S\s^\ 


Fig.  105 

therefore,  does  not  fall  to  zero  at  each  commutation, 
as  is  shown  in  Fig.  107,  but  the  current  curve  be- 
comes a  wavy  line  more  like  that  of  Fig.  108.  Picou 
says*  that  it  is  preferable  to  place  the  brushes  ahead 
of  the  point  of  least  sparking.  In  this  case  the  spark 
is  due  to  a  decreasing  current,  and  is  thin  and  weak. 

*  Machines  Dynamo-Electriqries,  p.  99. 


THE    MAGNETIC   CIRCUIT   OF   ALTERNATORS.     263 

With  the  brushes  behind  the  point  of  least  sparking, 
the  spark  is  clue  to  a  rising  current  and  it  is  of  great 
magnitude.  The  method  here  outlined  to  avoid  spark- 
ing does  not  seem  to  have  any  marked  advantages 
over  direct  commutation,  which  is  ordinarily  used  in 
American  self-exciting  machines.  The  advantage  of  a 
wavy  current  in  the  field,  instead  of  a  discontinuous 
one,  is  doubtless  equally  well  gained  in  the  American 


Fig.  106 


machines  by  the  use  of  copper  brushes  of  considerable 
thickness  on  the  rectifying  commutator,  which  short- 
circuit  the  supply  circuit  and  field  circuit  at  the  in- 
stant when  they  bridge  over  the  insulation  between  two 
segments.  In  composite-wound  machines  the  short- 
circuiting  of  the  series  field  supply,  which  occurs  for  an 
instant  at  each  commutation,  is  a  matter  of  no  mo- 


264  ALTERNATING   CURRENTS. 

ment,  since  cutting  the  small  resistance  of  the  series 
fields  in  and  out  of  the  main  circuit  cannot  have  an 
appreciable  effect  on  the  operation  of  the  machine. 
For  shunt-wound  self-exciting  machines,  the  current  for 
rectification  must  either  be  supplied  through  a  trans 


Fig.  107 

former  or  by  means  of  a  separate  exciting  coil  on  the 
armature,  to  avoid  disturbing  the  external  circuit  by 
short-circuiting  at  the  rectifier. 

For  the   purpose   of  varying  the  magnetizing  effect 
of  the  series  turns,  a  variable  shunt  is  often  connected 


Fig.  1O8 

across  their  terminals  (Fig.  109)  and  a  shunt  is  some- 
times placed  across  the  rectifier  terminals  in  such  a 
way  that  only  a  fixed  proportion  of  the  total  current 
is  rectified  and  passes  through  the  series  field  winding. 
This  is  for  the  purpose  of  reducing  the  difficulties 
caused  by  sparking,  by  reducing  the  current  to  be 
rectified. 


THE   MAGNETIC   CIRCUIT   OF   ALTERNATORS.      265 


266         ALTERNATING  CURRENTS. 


CHAPTER   VI. 

CHARACTERISTICS,    REGULATION,    ETC. 

72.  Alternator    Characteristics.  —  As   in    continuous 
current  machines  (Vol.  I.,  p.  195),  there  are  four  curves 
which  exhibit  particularly  important  relations  between 
the   functions    of    alternators.      These    curves,    which 
may  be  called  characteristics,   may  be  enumerated  as 
follows : 

1.  The  Curve  of  Magnetization. 

2.  The  External  Characteristics. 

3.  The  Loss  Line. 

4.  The   Magnetic   Distribution   Curve   and  Pressure 
Curve. 

73.  Curve  of  Magnetization.  —  The  curve  of  magneti- 
zation   shows   the   relation   between   the  total   electric 
pressure  developed   in  the  armature  and   the  ampere- 
turns  on  the  field.     From  the  total  electric  pressure  of 
the  armature,  the  value  of  Na  may  be  deduced  by  means 

of  the  formula  E  =  —    — %~*  provided  the  value  of  K  is 

known.  The  value  of  K  cannot  be  determined  exactly 
by  calculation,  but  may  be  ascertained  by  means  of  the 
fourth  curve.  The  experimental  determination  of  the 


CHARACTERISTICS,  REGULATION,  ETC.          267 

curve  of  magnetization  is  carried  out  exactly  as  in 
the  case  of  continuous-current  machines,  substituting 
for  the  plain  voltmeter  an  instrument  which  is  capable 
of  measuring  alternating  pressures.  It  is  desirable  that 
the  instrument  used  shall  indicate  the  effective  value  of 
the  pressure ;  hence,  the  measurements  must  be  made 
by  either  a  hot  wire  instrument  such  as  the  Cardew 
voltmeter,  an  electrostatic  instrument  modelled  after  the 
quadrant  electrometer,  or  a  non-inductive  form  of  high 
resistance  electrodynamometer.  All  instruments  used 
in  alternating-current  measurements  which  depend  for 
their  indications  upon  electrodynamic  action,  must  be 
constructed  with  no  masses  of  conducting  metal  about 
them,  or  their  constants  will  depend  upon  the  frequency 
of  the  current  measured.  This  is  due  to  the  dynamic 
effect  which  foucault  currents,  circulating  in  metallic 
masses,  must  have  on  the  currents  in  the  moving  parts 
of  the  instrument.  If  a  voltmeter  has  an  appreciable 
inductance,  its  reading  will  also  depend  upon  the  fre- 
quency, since  the  current  flowing  through  it  is  inversely 
proportional  to  ^2  +  4  Tr2/2^2.  From  this  it  is  readily 
seen  that  if  L  is  not  negligible  in  comparison  with  R, 
the  current  flowing  through  the  voltmeter  when  it  is 
connected  to  a  circuit,  and  hence  its  indication,  will  be 
dependent  upon  the  frequency.  The  indications  of  an 
inductive  voltmeter  will  always  be  less  when  it  is  con- 
nected to  an  alternating  circuit  than  when  it  is  con- 
nected to  a  continuous-current  circuit  of  equal  effective 
pressure.  The  self-inductance  of  electrodynamometers 
intended  for  use  as  amperemeters  is  usually  quite  small, 
but  in  some  cases  may  reach  a  millihenry.  Electrodyna- 


268  ALTERNATING   CURRENTS. 

mometers  which  are  intended  to  be  used  as  voltmeters, 
and  have  a  great  many  turns  of  wire  in  their  coils, 
sometimes  have  a  self-inductance  as  large  as  several 
hundredths  of  a  henry ;  but  the  commercial  voltmeters 
that  are  built  on  the  principle  of  electrodynamome- 
ters  have  a  considerable  non-inductive  resistance  in 
series  with  the  inductive  coils,  so  that  their  time  con- 
stant is  small. 

If  the  alternator  under  examination  was  designed  to 
be  a  self-exciting  one,  there  is  some  question  of  the 
comparative  magnetizing  effects  of  continuous  and  rec- 
tified currents.  In  general,  however,  as  we  have  already 
seen  (Sect.  71),  the  rectified  current  in  a  self-excited  field 
is  doubtless  always  a  wavy  one.  The  effective  value  of 
this,  as  indicated  by  an  electrodynamometer,  is  very 
nearly  the  same  as  the  average  value  indicated  by 
an  ordinary  amperemeter.  The  average  magnetizing 
effect  of  the  current  is  also  practically  equal  to  that  of 
a  continuous  current  which  gives  the  same  indications 
on  the  instruments.  A  wavy  current  tends  to  set  up 
foucault  currents  in  the  iron  of  the  magnetic  circuit 
and  thus  cause  heating,  but  this  result  is  not  marked. 

The  magnetizing  current  of  a  separately  excited  alter- 
nator may  be  caused  to  become  wavy  if  the  armature 
reactions  are  very  large.  It  has  already  been  shown 
that  the  effect  of  armature  reactions  is  a  periodic  one, 
and  when  the  periodic  effect  becomes  of  sufficient  mag- 
nitude it  causes  fluctuations  in  the  field  magnetism, 
which  react  upon  the  windings  and  throw  the  magnet- 
izing current  into  waves.  Curve  /,  Fig.  110,  shows  the 
current  curve  of  a  Stanley  arc-light  alternator  of  high 


CHARACTERISTICS,  REGULATION,  ETC. 


269 


self-inductance  when  the  machine  is  on  short  circuit. 
The  self-inductance  of  this  machine  is  so  great  that 
the  current  lag  is  nearly  90°  when  the  machine  is  short- 
circuited,  and  the  current  therefore  has  its  maximum 
value  when  the  centres  of  the  coils  are  almost  directly 
under  the  centres  of  the  pole  pieces.  The  armature 
current  therefore  has  a  maximum  effect  upon  the  field 


Fig.  110 

magnetism,  and  it  causes  such  a  variation  that  the  field 
current,  which  is  furnished  by  a  separate  continuous- 
current  dynamo,  is  thrown  into  waves  as  shown  by 
curve  //  of  the  figure.  The  relative  location  of  the 
poles  is  shown  in  the  figure,  and  the  forms  of  the  poles 
and  the  armature  teeth  with  their  relative  location  at 
the  instant  the  current  is  zero  are  shown  in  Fig.  in. 
Figures  112  and  113  show  the  same  features  when  the 
machine  is  worked  upon  a  full  load  of  40  arc  lamps,  in 


2/0 


ALTERNATING   CURRENTS. 


which  case  the  current  lag  is  not  quite  so  great.*  This 
effect  has  also  been  found,  but  to  a  less  degree,  in 
smooth-core  machines  with  surface  windings. 


Fig-,  in 


The  general  form  of  the  curve  of  magnetization  for 
an  alternator  is  similar  to  the  form  of  the  curve  for  a 


Pig.  112 


*  Tobey    and    Walbridge,    Stanley    Alternate-Current    Arc     Dynamo, 
Trans.  Amer.  Inst.  E.  £.,  Vol.  7,  p.  367. 


CHARACTERISTICS,  REGULATION,  ETC. 


2/1 


continuous-current    dynamo.     As  it  is  not  uncommon 
for  alternators  to  have  a  somewhat  larger  reluctance  in 


Fig-.  113 

the  air  space  than  have  continuous-current  dynamos  of 
the  same  size,  the  knee  in  the  alternator  curve  is  some- 


EXCITING  CURRENT 

Fig-.  114  a 


times  not  so  abrupt  as  it  is  in  the  case  of  continuous- 
current  machines  (Fig.  114  a  and  b).     For  studying  the 


272 


ALTERNATING  CURRENTS. 


details  of  the  design  of  the  magnetic  circuit,  the  curve 
may  be  resolved  into   component    curves    representing 


3,000 


2,000 


1,000 


'I- 


'12.8  AMP. 


5  10 

Fig-.  114  b 

Relation  of  pressure  to  exciting  current  with  different  currents  in 
armature. 

the  air  space,  frame,  and  armature,  exactly  as  explained 
in  Vol.  L,  p.  197. 

74.  External  Characteristic. — The  external  charac- 
teristic has  different  forms  which  depend  upon  the 
method  of  exciting  the  fields.  To  experimentally  deter- 
mine the  external  characteristic  of  an  alternator,  it  is 
excited  by  the  method  for  which  it  is  designed,  so  as  to 
give  its  normal  pressure  on  open  circuit.  The  volts  at 
its  terminals,  and  the  current  in  the  external  circuit, 


CHARACTERISTICS,  REGULATION,  ETC.          273 

are  measured  with  various  resistances  in  the  external 
circuit.  The  observations  may  be  plotted  in  a  curve, 
using  volts  as  ordinates  and  amperes  as  abscissas.  In 
separately  excited  alternators,  the  curve  cuts  the  verti- 
cal axis  at  the  highest  point,  and  then  gradually  falls ; 
the  decrease  of  the  ordinates  (drop  in  pressure)  being 
caused  by  the  effects  of  armature  resistance,  armature 
self-inductance,  and  armature  reactions.  The  measure- 
ments should  be  made  with  the  alternator  connected 
to  a  non-inductive  circuit,  since  the  magnitude  of  the 
armature  reactions  is  increased,  on  account  of  the 
greater  lag  of  the  current,  when  there  is  self-inductance 
in  the  external  circuit.  If  it  is  desired  to  quantitively 
determine  the  effect  of  lag  on  armature  reactions, 
curves  may  be  taken  with  different  values  of  induc- 
tance inserted  in  the  external  circuit.  The  portion  of 
the  drop  which  is  caused  by  self-inductance  and  arma- 
ture reactions  may  be  separated  from  that  caused  by 
resistance  by  the  formula,  E?  =  E?  -j-  Es2.  In  this  case 
Ei  is  the  open-circuit  pressure,  and  Ea  is  equal  to  the 
terminal  pressure,  for  the  load  C,  plus  CRa.  By  taking 
the  characteristics  of  an  alternator  when  worked  on 
circuits  of  different  known  resistances  and  self-induc- 
tances, the  effect  of  armature  reactions  may  be  deter- 
mined for  different  values  of  the  angle  of  lag.*  The 
self-inductance  of  some  alternators  is  so  great  that  the 
external  characteristic  droops  to  the  X  axis  at  a  current 
not  greatly  exceeding  full  load.  Figure  n$a  shows 
the  characteristics  of  two  alternators,  one  having  a 
small,  and  the  other  a  large,  self-inductance.  The  maxi- 

*  Compare  Kapp's  Dynamos,  Alternators,  arid  Transformers,  p.  383  et  seq. 
T 


274 


ALTERNATING   CURRENTS. 


mum  output  is  given  by  the  latter  when  the  armature 
current  has  the  value,  C=  -—-  A  — A,  according  to 

O-/LJ  jLa 

the  formula  on  page  237. 

The   external   characteristic  of   a  self-excited   shunt- 
wound  alternator  is  shown  in  Fig.  115$.     The  droop  in 


0  50  "100  150  200 

ARMATURE  CURRENT  IN  PER  CENT  OF  FULL  LOAD  CURRENT. 

Fig.  115  a 

the  curve  is  a  little  greater  than  it  would  be  for  the 
same  machine  separately  excited.  This  is  due  to  the  loss 
of  magnetizing  current  as  the  armature  losses  increase, 
and  the  terminal  pressure  decreases  accordingly  (com- 
pare Figs,  no  and  117,  Vol.  I.).  The  external  charac- 
teristic of  a  shunt-wound  self-excited  alternator  may  be 


CHARACTERISTICS,  REGULATION,  ETC.          275 


constructed  from  the  curve  of  magnetization  and  loss 
line,  by  the  process  that  has  already  been  explained  for 
continuous-current  dynamos  (Vol.  I.,  p.  205). 


75.  Loss  Line.  —  The  differences  between  the  ordi- 
nates  of  the  external  characteristic  of  a  separately 
excited  alternator  and  its  terminal  pressure  on  open 
circuit  are  the  ordinates  of  the  Loss  Line.  This  dif- 


2/6         ALTERNATING  CURRENTS. 

ference  in  the  ordinates  is  caused,  as  already  stated, 
by  the  loss  of  pressure  due  to  armature  resistance, 
counter  electric  pressure  due  to  self-induction,  and  the 
effect  of  armature  reactions.  Since  the  effect  of  arma- 
ture reactions  depends  upon  the  current  lag,  the  slope 
of  the  loss  line  is  dependent  on  the  kind  of  circuit 
upon  which  a  machine  is  worked,  and  the  slope  is  in- 
creased as  the  self-inductance  of  the  circuit  is  greater. 
When  the  alternator  has  a  small  armature  self-induct- 
ance and  is  worked  on  a  circuit  of  zero  or  negligible 
inductance,  the  loss  line  may  in  general  be  expected  to 
be  fairly  straight ;  since  the  drops  due  to  resistance,  in- 
ductance, and  armature  reactions  are  all,  within  narrow 
limits,  directly  proportional  to  the  armature  current, 
provided  the  armature  inductance  is  constant.  In  some 
machines  (especially  those  with  toothed  armatures),  these 
provisions  are  not  fulfilled  on  account  of  the  large  value 
of  the  inductive  pressure  which  is  in  quadrature  with 
the  active  pressure,  and  the  loss  line  may  be  consider- 
ably curved,  so  as  to  be  convex  towards  the  axis  of  cur- 
rent or  horizontal  axis.  The  amount  of  the  curvature 
is  dependent  upon  the  degree  of  saturation  to  which  the 
armature  core  is  subjected,  and  upon  the  effect  of  the 
air  space  in  the  circuit  of  the  magnetic  lines  which  are 
set  up  by  the  ampere-turns  of  the  armature.  Unless 
the  magnetic  induction  in  the  armature  is  denser  than 
other  conditions  will  admit  in  good  practice  (Sect.  61), 
the  self-inductance  is  not  likely  to  decrease  as  the  cur- 
rent rises.  Hence  the  loss  line  of  alternators  worked  on 
non-inductive  circuits  is  likely  to  be  nearly  straight,  or 
else  convex  towards  the  horizontal  axis  (Fig.  115  b). 


CHARACTERISTICS,  REGULATION,  ETC.          277 

76.  Instruments.  —  In  all  the  measurements  required 
in  determining  alternator  characteristics,  the  instru- 
ments which  are  used  must  be  carefully  selected  so  that 
they  may  properly  measure  alternating  currents.  Thus 
the  voltmeters  must  be  of  the  types  described  in  Sec- 
tion 73,  and  the  amperemeters  must  be  of  one  of  the 
following  types  : 

Electrodynamometers  or  Current  balances  which  have 
no  metal  in  their  frames  in  which  disturbing  foucault 
currents  may  be  set  up. 

Known  non-inductive  resistances  through  which  the 
current  to  be  measured  passes,  and  the  pressure  at  the 
terminals  of  which  may  be  measured  by  means  of  an 
electrostatic  or  hot  wire  voltmeter,  or  by  means  of  an 
electrodynamic  voltmeter  having  a  negligible  time  con- 
stant. 

Or  finally,  amperemeters  dependent  for  their  indica- 
tions upon  the  heating  effect  of  the  current. 

These  instruments  may  be  standardized  and  calibrated 
in  the  usual  manner  with  continuous  currents,  after 
which  they  will  give  correct  indications  of  the  effective 
values  of  alternating  currents  for  any  frequency  within 
the  usual  commercial  limits.  If  other  types  of  ampere- 
meters are  used,  they  must  be  calibrated  by  comparison 
with  instruments  of  one  of  the  types  described  above, 
using  an  alternating  current,  for  the  calibrating,  which 
has  exactly  the  same  frequency  as  that  which  is  to  be 
measured  in  the  test.  Amperemeters  which  depend 
for  their  indications  upon  the  attraction  of  a  solenoid 
upon  a  laminated  iron  core,  are  likely  to  give  widely 
different  indications  for  equal  currents  of  different  fre- 


ALTERNATING    CURRENTS. 

quencies.      This    is    principally   due    to    the    effects    of 
foucault  currents  and  hysteresis. 

77.  The  Magnetic  Distribution  Curve,  and  Curves  of 
Pressure  and  Current. — These  curves  may  be  experi- 
mentally determined  by  various  methods.  They  consist 
of  a  series  of  curves  which  are  closely  interrelated,  but 
may  be  and  in  fact  are  likely  to  be,  of  quite  dissimilar 
forms.  The  form  of  the  curve  representing  the  wave 
of  impressed  pressure,  or  total  pressure  developed  in 
the  armature  of  an  alternator,  is  directly  dependent 
upon  the  distribution  of  the  magnetism  over  the  pole 
faces,  and  also  on  the  arrangement  of  the  armature 
windings.  By  poor  designing,  either  of  these  may  be 
given  a  controlling  influence  to  the  exclusion  of  the 
other.  The  two-pole  machine  with  Gramme  armature, 
discussed  in  Section  5,  gives  an  excellent  instance.  The 
differential  action,  which  occurs  in  the  coils  of  this  arma- 
ture, makes  its  curve  of  pressure  almost  independent  of 
the  distribution  of  the  magnetism  over  the  pole  faces, 
provided  the  same  total  number  of  lines  of  force  is  cut 
by  the  conductors  per  revolution  ;  and  the  maximum 
value  of  the  pressure  is  entirely  independent  of  the 
magnetic  distribution.  This  is  shown  by  Fig.  116, 
where  the  full  line  in  one  cut  shows  the  curve  of  press- 
ure developed  in  the  armature  with  a  uniform  distribu- 
tion of  the  field,  and  the  full  line  in  the  other  shows 
the  curve  of  pressure  with  the  same  total  field  greatly 
distorted.  The  dotted  lines  in  the  cuts  show  the  distri- 
bution of  the  magnetism  over  the  pole  faces.  The  con- 
tinuous pressure  developed  in  the  armature  when  the 
machine  is  operated  as  a  continuous-current  dynamo, 


CHARACTERISTICS,  REGULATION,  ETC.         279 

is  not  affected  by  the  distortion,  provided  the  brushes 
are  always  placed  on  the  neutral  plane  and  the  total 
magnetism  passing  through  the  armature  remains  con- 
stant. When  the  machine  is  converted  into  an  alter- 
nator by  the  addition  of  collecting  rings,  the  maximum 
instantaneous  pressure  is  equal  to  the  pressure  devel- 
oped in  the  continuous-current  machine,  and  is  therefore 
independent  of  the  distribution  of  the  magnetism,  but 
the  form  of  the  curve  representing  the  wave  of  pressure 
is  slightly  altered,  as  shown  in  Fig.  116.  Now  sup- 


Fig-.  116 

pose  the  same  machine  to  be  arranged  with  a  single 
narrow  coil  on  the  armature.  The  change  in  the  mag- 
netic distribution  now  not  only  changes  the  form  of  the 
pressure  curve  proportionally,  but  also  changes  in  a 
marked  manner  the  maximum  value  of  the  instantaneous 
pressure.  The  difference  in  the  curves  of  pressure  de- 
veloped by  the  broad  and  narrow  coil  armatures  is  due 
to  the  effect  of  differential  action  in  the  broad  coil  arma- 
ture. It  has  already  been  shown  that  differential  action 
occurs  to  some  degree  in  commercial  alternators,  but  it 
does  not  occur  to  a  sufficient  degree  to  make  the  form 


280  ALTERNATING    CURRENTS. 

of  the  pressure  wave  independent  of  the  magnetic  dis- 
tribution. It  is  therefore  true  that  the  magnetic  distri- 
bution largely  influences  the  form  of  the  pressure  wave, 
and  the  distribution  should  therefore  be  carefully  studied 
during  the  development  of  a  type  of  alternators.  A 
proper  study  of  the  magnetic  distribution  and  of  the 
arrangement  of  the  armature  windings,  makes  it  pos- 
sible to  so  design  an  alternator  that  it  will  produce 
any  desired  form  of  pressure  wave.  This  is  an  important 
point  which  will  receive  additional  attention  later. 

The  angular  relation  between  the  curves  representing 
the  magnetic  distribution  and  the  impressed  pressure,  is 
interesting.  The  ordinate  of  the  curve  of  pressure  at 
any  point,  is  proportional  to  the  rate  at  which  lines  of 
force  are  cut  by  the  armature  conductors  at  that  point, 
the  rate  being  taken  algebraically.  Consequently,  the 
pressure  is  zero  when  the  magnetization  is  all  symmetri- 
cally threaded  through  the  coils ;  that  is,  when  the  alge- 
braic rate  of  cutting  lines  by  the  conductors  is  zero. 

The  pressure  is  a  maximum 
when  the  rate  of  cutting 
lines  is  the  greatest ;  that 
is,  when  the  algebraic  sum- 
mation of  the  number  of 
lines  threaded  through  the 
coils  is  a  minimum.  A 
curve  which  shows  the 

algebraic  summation  of  the 

Flgr*  117  number    of    lines    threaded 

through  the  coils  at  each  instant,  therefore,  has  an 
angular  position  which  is  90°  from  that  of  the  pressure 


CHARACTERISTICS,  REGULATION,  ETC.          281 

curve  (Fig.  117).  The  form  and  dimensions  of  this 
curve  evidently  depend  upon  the  actual  distribution  of 
the  magnetism  and  the  arrangement  of  the  armature 
windings.  When  the  pressure  curve  is  irregular,  and 
the  angular  relation  is  not  made  evident  from  the  curve, 
as  in  Fig.  118,  the  point  at  which  the  curve  of  mag- 
netism cuts  the  X-axis  may  yet  be  easily  found,  since  it 
is  directly  under  the  centre  of  gravity  of  the  pressure 
curve.  That  is,  since  as  many  lines  of  force  must  be 
withdrawn  from  the  coils  as  are  inserted,  for  each  loop 
of  the  curve,  the  summation  of  the  ordinates  of  the 
pressure  curve  on  each  side  of  the  crossing  must  be 
equal. 

This  curve,  which  shows  the  algebraic  number  of 
lines  of  force  which  are  threaded  at  each  instant  through 
the  coils,  may  be  easily  deduced  from  the  curve  of 
pressure.  Erect  an  ordinate  to  the  pressure  curve 
which  bisects  the  area ;  then  by  means  of  ordinates 
divide  the  half  areas  into  a  number  of  small  areas.  The 
magnetism  threaded  through  the  armature  coils  is  alge- 
braically equal  to  zero  at  the  instant  represented  by  the 
bisecting  ordinate,  and  the  algebraic  value  of  the  mag- 
netism threaded  through  the  armature  coils  at  any 
other  instant  is  proportional  to  the  area  enclosed  by 
the  pressure  curve  between  the  corresponding  ordinate 

and  the  bisecting  ordinate,  since  e  =  —  and  N  =*Ledt. 

at 

Therefore,  the  ordinates  of  the  curve  representing  the 
magnetism  threaded  through  the  coils  are,  at  the  in- 
stants represented  by  the  ordinates  which  divide  the 
pressure  curve  into  small  areas,  proportional  to  the  area 


282 


ALTERNATING  CURRENTS. 


between  the  corresponding  instantaneous  pressure  orcli- 
nate  and  the  bisecting  ordinate.  The  full  process  is, 
therefore,  as  follows  :  lay  off  on  each  ordinate  a  length 
from  the  Jf-axis  proportional  to  the  area  enclosed  by 
the  pressure  curve  between  the  respective  ordinate 
and  the  bisecting  ordinate.  The  points  thus  found  are 
points  on  the  desired  curve.  It  is  evident  that  the 
maximum  ordinate  of  the  curve  comes  at  the  instant 


when  E  is  equal  to  zero.  The  length  of  this  ordinate 
is  equal  to  Na,  and  the  scale  of  the  curve  may  thus  be 
conveniently  fixed.  The  curve  may  not  be  symmetrical, 
but,  with  a  fixed  value  of  Na  and  a  fixed  armature  wind- 
ing, the  successive  loops  must  always  be  exactly  alike, 
though  they  may  be  looked  upon  as  alternately  posi- 
tive and  negative,  since  the  magnetism  is  alternately 
threaded  through  the  coils  in  opposite  directions.  The 
corresponding  curves  of  pressure  and  magnetism  for 


CHARACTERISTICS,  REGULATION,  ETC. 


283 


Fig.  119 


various  forms  of  pressure  curves  are  shown  in  the 
accompanying  figures.  The  construction  of  the  second 
curve  from  the  first  is 
shown  by  the  dotted 
lines.  In  Fig.  118  the 
pressure  curve  is  one 
experimentally  deter- 
mined from  a  Stanley 
arc  light  alternator 
when  working  on  a 
full  load  of  arc  lights.* 
In  Fig.  119  the  press- 
ure curve  is  an  equi- 
lateral triangle,  in  Fig. 
1  20  it  is  a  sinusoid, 
and  in  Fig.  121  it  is  a 
rectangle.  The  press- 
ure curves  of  Figs.  122 
and  123  are  respec- 
tively a  flat-topped 
curve  and  a  parabola.f 
Since  the  electrical 
pressure  is  proper- 
tional  to  the  rate  of  change  of  the  number  of  lines  of 
force  threaded  through  the  armature  coils,  the  ordinates 
of  the  pressure  curve  are  proportional  to  the  tangents 
of  the  curve  representing  the  number  of  lines  enclosed 

*  Tobey    and    Walbridge,    Stanley    Alternate-Current    Arc    Dynamo, 
Trans.  Amer.  Inst.  E.  £.,  Vol.  7,  p.  367. 

t  Emery,  Alternating    Current    Curves,   Trans.    Amer.    Inst.   E.    £.y 
Vol.  12,  p.  433. 


Fig.  12O 


121 


284 


ALTERNATING   CURRENTS. 


by  the  armature  coils.     Figure   124  shows  a  graphical 
construction  for  determining  the   pressure   curve  from 


Fig.  122 


Pig-.  123 


this  magnetic  curve.  Oa\  Ob',  and  OA  are,  by  construc- 
tion, proportional  to  the  tangents  of  the  angles  with  the 
Jf-axis  made  by  the  tangents  to  the  magnetic  curve  at 
/!,  /2,  and  O.  O'  is  any  point,  and  O'a',  O'b!,  O'A  are 
drawn  parallel  respectively  to  aa,  bb,  and  the  tangent 


at  O.  The  points  of  intersection  of  horizontals  drawn 
from  a',  b' ,  etc.,  and  verticals  drawn  from  pv  /2,  etc., 
are  points  on  the  required  pressure  curve. 


CHARACTERISTICS,  REGULATION,  ETC. 


285 


A  more  directly  useful  magnetic  curve  is  one  showing 
the  distribution  of  the  lines  of  force  over  the  pole  faces. 
This  curve  is  analogous  to  the  magnetic  distribution 
curve  of  continuous-current  machines  (Vol.  I.,  p.  208). 
It  may  be  experimentally  determined  by  the  fourth  or 
test-coil  method  given  in  Vol.  I.,  p.  211.  It  is  prob- 
able that  the  magnitude  of  the  periodic  effect  of  arma- 
ture reactions  may  also  be  experimentally  determined 
by  using  two  test  coils,  one  of  which  is  placed  in  a  posi- 
tion coincident  with  the  armature  coils,  and  the  other  of 
which  is  placed  so  that  its  phase  is  90°  in  advance.  If 


Fig.  125 

the  average  distribution  of  magnetism  over  the  pole 
faces  of  a  machine  is  known,  it  is  evidently  possible  to 
approximately  determine  the  form  of  the  pressure  curve 
which  will  be  produced  by  any  particular  arrangement 
of  the  windings.  It  is  also  equally  possible  to  determine 
the  arrangement  of  the  windings  required  to  give  any 
desired  form  of  pressure  curve.  Again,  if  a  particular 
form  of  winding  is  desirable,  the  magnetic  distribution 
which  is  necessary  to  give  a  desired  pressure  curve  may 
be  determined.  This  distribution  may  then  be  used  as 
a  guide  in  designing  the  width  and  shape  of  the  pole 


286  ALTERNATING   CURRENTS. 

faces.  The  application  of  the  magnetic  distribution 
curve  is  illustrated  in  Fig.  125.  The  dimensions  and 
form  of  the  pole  pieces  and  of  an  armature  coil  belong- 
ing to  an  alternator,  are  indicated  in  the  figure.  The 
ordinates  of  the  line  ABCD  represent  the  magnetic 
density  in  the  air  space.  When  the  coil  is  in  the 
instantaneous  position  represented,  the  value  of  the  elec- 
tric pressure  is  zero.  As  the  coil  moves,  each  con- 
ductor cuts  lines  of  force.  Suppose  that  in  one  twenty- 
fourth  of  a  period  the  coil  has  moved  from  the  position 
indicated  by  the  letters  x,  y,  to  x' ,  y'.  During  this  mo- 
tion each  conductor  has  cut  a  certain  number  of  lines  of 
force,  and  the  number  cut  by  all  the  conductors  is 
approximately  proportional  to  the  sum  of  the  areas  of 
the  curve  ABCD  taken  from  x  to  x1  and  from  y  to  y' . 
The  shorter  the  step  taken,  the  more  accurate  this 
becomes.  The  average  pressure  developed  during  this 
interval  is  also  proportional  to  the  same  area.  Con- 
sequently an  ordinate  which  is  numerically  equal  to 
the  area,  may  be  erected  at  the  point  a  —  7-|-°  (one 
forty-eighth  of  a  cycle)  to  approximately  represent  the 
pressure  at  that  point  in  the  revolution  of  the  arma- 
ture. This  proceeding  may  be  repeated  through  the 
half  period  taking  the  algebraic  summation  of  the 
.pressures  developed  in  the  two  halves  of  the  coil,  and 
the  outline  of  the  pressure  curve  is  thus  determined.* 

The  curve  representing  the  current  wave  also  repre- 
sents, when  taken  to  the  proper  scale,  the  curve  of 
active  electric  pressure.  From  preceding  pages  it  is 
evident  that  the  curves  of  active  and  impressed  press- 

*  Kapp's  Dynamos y  Alternators,  and  Transformers,  p.  364  et  seq. 


CHARACTERISTICS,  REGULATION,  ETC.          287 

ures  have  the  same  .forms  and  are  superposed,  if  the  cir- 
cuit in  which  they  act  is  non-inductive  and  without 
capacity.  They  have  the  same  form,  also,  when  the  cir- 
cuit is  inductive,  if  the  impressed  pressure  is  sinusoidal, 
provided  the  inductance  is  independent  of  the  instan- 
taneous value  of  the  current  and  the  armature  reactions 
are  approximately  uniform.  The  condition  of  a  uniform 
inductance  can  only  hold  when  no  iron  is  enclosed  in 
any  portion  of  the  circuit.  In  general  this  condition 
is  not  found  in  commercial  service.  Even  with  a  uni- 
form inductance  the  curves  of  impressed  and  active 
pressures  will  not  coincide  in  phase,  since  phase  coinci- 
dence between  them  can  occur  only  when  the  circuit  is 
without  either  inductance  or  capacity,  or  these  exactly 
neutralize  each  other.  As  the  latter  is  also  a  condi- 
tion not  often  found  in  commercial  service,  it  may  be 
said  that  in  general  curves  of  impressed  and  active 
pressures  are  neither  similar  in  form  nor  coincident 
in  phase;  but  they  are  always,  perforce,  of  the  same 
frequency. 

The  curves  are  usually  plotted  to  rectangular  coor- 
dinates; inductions,  pressures,  or  instantaneous  cur- 
rents being  plotted  as  ordinates,  and  angular  degrees 
or  time  as  abscissas.  To  more  definitely  locate  the 
phases  it  is  not  unusual  to  indicate  the  position  of  the 
pole  pieces  by  laying  them  off  at  the  top  or  the  bot- 
tom of  the  plot  (Fig.  126.)*  No  systematic  study  has 
been  made  of  the  distribution  of  magnetism  over  the 
pole  faces  of  alternators.  Such  a  study,  as  already 

*  Compare  Trans.  Amer.  Inst.  E.  E.,  Vol.  7,  pp.  i,  311,  324,  367; 
also  London  Electrician,  Vol.  28,  p.  90;  and  Elect.  World,  Vol.  18,  p.  368. 


288         ALTERNATING  CURRENTS. 

intimated,  would  be  of  much  value  in  determining  the 
most  satisfactory  form  for  pole  pieces  and  the  arrange- 
ment for  armature  windings.  To  make  the  work  com- 
plete, it  should  cover  alternators  with  various  types 
of  armatures  when  worked  at  various  loads  under  dif- 
ferent conditions  of  current  lag.  Since,  at  any  in- 
stant, the  effect  of  the  armature  current  on  the  magne- 
tization of  the  pole  pieces,  depends  upon  the  position  of 
the  armature  coils  as  well  as  the  strength  of  the  cur- 
rent, this  effect  is  evidently  a  variable,  and  consequently 


Fig.  126 

the  distribution  of  the  magnetism  will  not  be  constant. 
In  other  words,  both  the  cross  turns  and  back  turns  for 
any  load  vary  continuously  during  each  period,  and 
therefore  the  magnetic  distribution  varies  with  the  posi- 
tion of  the  armature.  The  magnitude  of  the  variation 
of  the  magnetic  distribution  has  never  been  fully  deter- 
mined. It  probably  is  not  very  great  in  machines  with 
smooth-core  armatures,  and  an  average  distribution  may 
be  assumed  as  satisfactorily  representing  working  con- 
ditions. In  machines  with  toothed  or  pole  armatures, 
the  effect  of  armature  reactions,  and  the  movement  of 


CHARACTERISTICS,  REGULATION,  ETC.          289 

the  teeth  across  the  pole  faces,  is  often  sufficient  to 
cause  regular  pulsations  in  the  field  magnetism  and 
extremely  marked  distortions  in  the  pressure  curve. 
The  pulsations  are  sometimes  sufficient  to  materially 
affect  the  magnetizing  current.  Figure  112,  curve  //, 
shows  an  experimentally  determined  curve  of  the  field 
current  of  an  alternator  having  a  toothed  armature  core 
and  a  large  self-inductance  in  the  armature.  The 
machine  was  excited  by  a  small  shunt-wound  exciter.* 

The  most  satisfactory  method  of  studying  the  mag- 
netic distribution  is  by  means  of  a  test  wire  laid  on  the 
surface  of  the  armature.  This  is  similar  to  the  method 
by  test  coil  on  armature  explained  on  p.  211  of  Vol.  I. 
To  study  the  effect  of  the  armature  upon  the  magnetic 
distribution,  the  test  wire  must  be  successively  located 
at  different  points  on  the  armature  within  an  angular 
space  equal  to  the  pitch  of  the  poles. 

78.  Methods  of  tracing  Pressure  and  Current  Waves. 
—Various  methods  may  be  used  for  experimentally  deter- 
mining the  form  of  electric  pressure  or  current  curves. 

i.  Ballistic  Method.  A  ballistic  galvanometer  is 
attached  to  the  terminals  of  the  alternator  to  be 
tested.  The  fields  are  excited  in  the  usual  manner 
to  the  degree  desired.  The  armature  is  then  quickly 
advanced  through  a  small  arc  of  revolution.  The 
throw  of  the  galvanometer  is  read.  The  armature  is 
again  advanced  through  an  equal  arc  and  the  throw 
read.  This  is  continued  until  the  armature  has  been 
advanced  through  a  distance  equal  to  twice  the  pitch, 
when  one  complete  period  of  the  pressure  curve  will 

*  See  Trans.  Amer.  Inst.  E.  E.,  Vol.  7,  p.  374. 
U 


290         ALTERNATING  CURRENTS. 

have  been  completed.  Plotting  the  galvanometer  indi- 
cations as  ordinates  corresponding  with  the  angular 
advance,  the  pitch  being  taken  as  180°,  gives  the  curve 
of  pressure  when  no  current  is  flowing  in  the  armature. 
This  method,  therefore,  gives  an  opportunity  to  study 
the  effect  of  armature  reactions  by  comparison  with 
curves  taken  by  later  methods. 

This  method  may  be  modified  by  reversing  the  field 
current  when  the  armature  is  in  the  successive  posi- 
tions, instead  of  quickly  moving  the  armature.  The 
throws  of  the  galvanometer,  which  are  thus  given,  are 
proportional  to  the  number  of  lines  of  force  which  pass 
through  the  armature  coils  when  the  armature  is  in 
the  corresponding  positions.  The  experimentally  deter- 
mined curve  therefore  shows  the  number  of  lines  of 
force  which  pass  through  the  armature  coils  at  each 
point,  and  from  this  curve  the  form  of  the  pressure 
curve  is  easily  derived,  as  already  explained  (page  284). 
Figure  124  shows  the  curves  thus  determined. 

2.  Gerard's  Method.  This  is  another  method  by 
which  the  curve  of  pressure  of  an  alternator,  when  no 
current  flows  in  the  armature,  may  be  traced  without 
special  apparatus.  The  alternator  to  be  tested  is  ro- 
tated at  a  very  slow  speed,  the  field  being  excited  in  the 
usual  manner.  The  terminals  of  the  machine  to  be 
examined  are  connected  to  a  shunted  d'Arsonval  gal- 
vanometer. The  natural  rate  of  oscillation  of  the  galva- 
nometer bobbin  is  made  quite  rapid,  as  compared  with 
the  period  of  the  pressure  supplied  by  the  alternator 
at  its  slow  speed.  Then  the  deflection  of  the  needle 
at  each  instant  will  be  proportional  to  the  instantane- 


.CHARACTERISTICS,  REGULATION,  ETC.          291 

ous  pressure.  By  moving  a  sheet  of  sensitized  paper 
before  the  galvanometer  mirror,  which  throws  upon  it 
a  beam  of  light,  the  curve  of  pressure  may  be  perma- 
nently recorded.* 

Each  of  the  following  methods  requires  the  use  of  a 
revolving  contact  maker  of  some  kind,  and  they  there- 
fore have  much  in  common.  The  principal  differences 
in  the  methods  relate  to  the  type  of  instruments  used 
to  give  the  indications,  and  the  convenience  with  which 
the  manipulations  may  be  made.  Whether  current  or 
pressure  curves  are  to  be  obtained,  instantaneous  press- 
ure measurements,  only,  are  made.  For  the  former,  the 
instantaneous  pressures  are  taken  at  the  terminals  of 
a  non-inductive  resistance,  and  the  instantaneous  cur- 
rents are  readily  deduced. 

3.  Jouberfs  Method  (1880).  The  terminals  of  the 
alternator  armature,  or  of  one  bobbin  of  the  arma- 
ture, are  connected  to  a  condenser  in  the  following 
manner :  One  armature  terminal  is  connected  perma- 
nently to  one  terminal  of  the  condenser ;  the  other  ar- 
mature terminal  is  connected  to  a  rotating  point  which 
may  be  put  in  connection  with  the  free  terminal  of  the 
condenser,  when  the  armature  is  at  any  desired  point  in 
its  rotation.  At  the  instant  this  contact  is  made,  the 
condenser  receives  a  charge  which  is  proportional  to  the 
instantaneous  pressure  in  the  armature,  and  which  may 
be  measured  by  discharging  the  condenser  through  a 
ballistic  galvanometer.  By  setting  the  contact  to  cor- 
respond with  various  points  in  the  revolution  of  the 
armature,  the  corresponding  instantaneous  pressures 

*  See  Gerard's  Lemons  sur  l'£lectricite,  Vol.  I.,  p.  565,  3d  ed. 


2Q2  ALTERNATING    CURRENTS. 

may  thus  be  measured  and  the  curve  of  pressure  may 
be  plotted  (Fig.  127).  The  contact  maker  used  by 
Joubert  was  an  insulated  pin  set  in  the  armature  shaft, 
against  which  a  brush  could  be  made  to  bear  at  any 
point  in  the  revolution,  as  explained  on  p.  211,  Vol.  I. 
A  quadrant  electrometer  may  be  used  in  place  of  the 
condenser  and  ballistic  galvanometer ;  in  which  case  it 


Pig-.  127 

is  desirable  to  introduce  a  condenser  permanently  in 
parallel  with  the  electrometer,  to  neutralize  the  effect 
of  leakage  in  the  test  circuit.* 

Joubert's  investigations  made  in  1880  resulted  in  the 
first  determination  of  the  curve  of  pressure  of  an  alter- 
nator (see  page  28).  The  investigations  of  Duncan, 
Hutchinson,  and  Wilkes  probably  produced  the  earliest 
series  of  experimental  curves  showing  the  relations 
between  the  waves  of  pressure  and  current  in  circuits 
of  different  kinds.  The  investigations  of  Searing  and 

*  See  Joubert,  Comptes  Rendus,  Vol.  91,  1880,  p.  161;  Joubert,  Journal 
de  Physique,  1881;  Duncan,  Hutchinson,  and  Wilkes,  Electrical  World, 
Vol.  n,  1888,  p.  1 60;  Searing  and  Hoffman,  Jour.  Franklin  Institute, 
Vol.  128,  1889,  p.  9*3;  Ryan,  Trans.  Amer.  hist.  E.  E.,  Vol.  7,  1890, 
p.  i;  Tobey  and  Walbridge,  Trans.  Amer.  Inst.  E.  E.,  Vol.  7,  p.  367; 
Blondel,  La  Lumiere  Electrique,  Vol.  41,  pp.  401  and  507;  Hopkinson's 
Dynamo  Machinery  and  Allied  Subjects,  p.  187;  etc. 


CHARACTERISTICS,  REGULATION,  ETC. 


293 


Hoffman  were  the  first  made  upon  an  alternator  with 
iron  in  the  armature  core.  Their  results  showed  the 
curve  of  pressure  developed  in  a  smooth-core  drum 
armature  to  approach  a  sinusoid  (Fig.  128). 


•i    i    i 


ALTERNATOR  CURVE. 
SINE  CURVE. 
PARABOLA. 


15 


30  35 

Fig.  128 


45 


50 


4.  Ryaris  Method  (1889).  Professor  Ryan  of  Cornell 
University,  in  conjunction  with  Professor  Merritt, 
carried  out  a  series  of  investigations  in  1889,  in  which 
an  entirely  different  and  original  arrangement  of  the 
measuring  instruments  was  used.  The  use  of  a  con- 


294 


ALTERNATING   CURRENTS. 


denser  and  ballistic  galvanometer  in  determining  points 
of  the  pressure  and  current  curves  involves  long  and 
laborious  manipulation,  and  introduces  various  elements 
of  inaccuracy.  On  the  other  hand,  a  quadrant  elec- 
trometer is  fairly  re- 
liable, is  direct  read- 
ing, and  is  approxi- 
mately dead  beat,  but 
has  a  limited  range. 
A  satisfactory  elec- 
trometer for  use  in 
a  long  investigation 
of  the  kind  under 
consideration  should 
have  a  wide  range, 
throughout  which  the 
indications  should  be 
uniformly  accurate. 
It  is  also  desirable 
that  the  instrument 
have  a  simple  law  and 
an  invariable  con- 
stant. To  fulfil  these 


Fig.  129 


conditions,  Professor  Ryan  designed  a  special  zero  read- 
ing electrometer  which  consists  essentially  of  a  cylin- 
drical electrometer  needle  A,  and  four  quadrants  Q,  Q 
(Fig.  129).  On  the  upper  side  of  the  electrometer 
needle  is  hung  a  magnetized  steel  mirror  C,  which 
serves  both  as  a  magnetic  needle  and  as  a  mirror.  The 
needle  is  suspended  by  a  silk  fibre,  and  is  put  in  me- 
tallic contact  with  the  case  by  means  of  a  loop  of  very 


CHARACTERISTICS,  REGULATION,  ETC.         295 

fine  silver  wire  S.  The  electrometer  case  is  circular, 
and  the  magnetic  needle  is  arranged  to  be  in  its  centre. 
Around  the  case  is  wound  a  coil  BB  of  fine  insulated 
wire.  When  the  plane  of  this  coil  stands  in  the  mag- 
netic meridian,  the  coils  and  magnetic  needle  make  a 
tangent  galvanometer.  On  the  other  hand,  when  the 
electrometer  needle  is  connected  to  the  case  and  one 
pair  of  quadrants,  it  makes,  in  combination  with  the 
other  pair  of  quadrants,  a  quadrant  electrometer  con- 
nected for  idiostatic  use.  If  the  terminals  of  the  elec- 
trometer are  connected  to  a  circuit,  the  pressure  of 
which  is  to  be  measured,  the  needle  experiences  a  de- 
flecting couple  which  is  proportional  to  the  square  of 
the  pressure.  For,  we  have  seen  in  Vol.  I.,  p.  19,  that 
the  attractive  force  between  two  charged  plates  is 


r1  = 


where  V  is  the  difference  of  potential  between  the 
plates,  A  is  their  area,  and  D  is  their  distance  apart. 
In  this  case  both  A  and  D  are  unknown.  Their  effec- 
tive values  are  constant,  however,  since  the  instrument 
is  designed  to  be  used  as  a  zero  instrument ;  that  is,  the 
needle  always  has  a  fixed  position  with  respect  to  the 
quadrants  when  its  indications  are  read.  To  hold  the 
needle  at  zero  when  the  needle  and  quadrants  are 
charged,  a  current  is  passed  through  the  coils  BB  in 
such  a  direction  and  of  such  a  strength  that  its  deflect- 
ing couple  on  the  magnetized  mirror,  is  opposite  and 
equal  to  the  couple  exerted  on  the  electrometer  needle 
by  the  pressure  which  is  to  be  measured.  The  latter  is 


296         ALTERNATING  CURRENTS. 

then  proportional  to  the  square  root  of  the  balancing  cur- 
rent, since  the  deflecting  couple  due  to  a  circular  current 
is  directly  proportional  to  the  current.  The  electrometer 
may  be  calibrated  by  finding  the  currents  flowing  in  the 
coils  which  are  required  to  balance  known  pressures,  and 
a  curve  of  calibration,  which  should  be  a  parabolic  line, 
may  be  plotted.  Instead  of  using  a  galvanometer  for 
measuring  the  current  in  the  balancing  coils,  a  cell  of 
constant  pressure  and  of  low  resistance  may  be  used  to 
furnish  current  to  the  coils  which  are  connected  in  series 
with  a  variable  resistance.  Then  the  current  flowing 
is  inversely  proportional  to  the  total  resistance  in  the 
circuit  of  the  coils  and  cells,  and  therefore  the  electric 
pressure  between  the  electrometer  terminals  is  inversely 
proportional  to  the  square  root  of  the  resistance.* 

5.  Mershoris  Method.  A  galvanometer  with  a  suffi- 
ciently great  time  of  vibration  will  be  steadily  deflected 
by  the  succession  of  impulses  which  it  receives  when 
connected  in  circuit  with  a  contact  maker.  This  deflec- 
tion may  be  balanced  by  a  steady  electric  pressure 
which  is  introduced  in  the  circuit  in  series  with  the 
galvanometer  and  contact  maker  (Fig.  130).  When 
the  balancing  pressure  reduces  the  galvanometer  deflec- 
tion to  exactly  zero,  the  balancing  pressure  is  evidently 
equal  to  the  instantaneous  pressure  at  the  contact 
maker.  This  arrangement  of  the  apparatus  unfort- 
unately lacks  sensitiveness  when  used  in  measuring 
pressures  which  have  a  wide  range  of  values.  To 
correct  this  fault,  Mr.  R.  D.  Mershon,  of  the  Westing- 

*  Trans.  Amer.  Inst.  E.  E.,  Vol.  7,  p.  I. 


CHARACTERISTICS,   REGULATION,    ETC.         297 


house  Electric  Company,  replaced  the  galvanometer 
by  a  telephone  receiver  (Fig.  131).  Whenever  contact 
is  made  by  the  contact  maker,  a  sharp  click  is  heard 
in  the  telephone,  unless  a  balance  of  pressure  exists. 

C.M. 


Fig.  ISO 


In  order  to  get  the  balance  with  great  exactness,  it  is 
usually  well  to  find  the  value  of  the  balancing  pressure 
when  it  is  increased  from  a  smaller  value,  and  also  when 
it  is  decreased  from  a  larger  value.  The  mean  value 


CONTACT-MAKER 

REVERSING  KEY 

EPHONE   RECEIVER 


Fig.  131 

given  by  the  two  balancing  points  may  be  taken  to 
represent  the  true  balance.  It  is  best  to  place  a  con- 
denser in  parallel  with  either  the  galvanometer  or  tele- 
phone when  this  arrangement  is  used.* 

*  Electrical  World,  Vol.  18,  p.  140;  Hopkinson's  Dynamo  Machinery 
and  Allied  Subjects,  p.  189.  This  has  been  modified  by  Duncan  for  espe- 
cially accurate  work  {Electrical  Engineer,  Vol.  19,  p.  192). 


298 


ALTERNATING   CURRENTS. 


6.  Duncan  s  Method  (1891).  It  is  frequently  desira- 
ble to  make  simultaneous  determinations  of  several 
pressure  and  current  curves.  In  this  case,  if  one  of 
the  methods  is  used  in  which  the  indications  are  gained 
by  the  intervention  of  either  a  condenser  or  an  electrom- 
eter, a  contact  maker  is  required  for  each  curve.  Dr. 
Louis  Duncan  of  Johns  Hopkins  University,  assisted 
by  Mr.  Carichoff  and  others,  devised  a  method  which 
avoids  this  multiplication  of  contact  makers.  The  read- 
ings are  made  upon  special  electrodynamometers.  One 


Fig.  132 

of  these  is  provided  for  each  curve  which  is  to  be 
traced,  and  the  fixed  coil  of  each  is  connected  to  the 
circuit  to  which  its  curve  belongs.  The  movable  coils 
are  all  wound  alike  of  fine  wire,  and  are  connected  in 
series.  In  circuit  with  them  are  connected  a  few  cells 
of  storage  battery  and  a  contact  maker  (Fig.  132). 

It  is  evident  that  if  alternating  currents  are  passed 
through  the  fixed  coils  of  the  electrodynamometers  and 
at  a  certain  moment  an  instantaneous  current  be  passed 
through  the  movable  coils,  each  will  receive  an  im- 
pulse that  is  proportional  to  the  instantaneous  value 


CHARACTERISTICS,   REGULATION,   ETC.         299 


of  the  alternating  current  in  its  fixed  coil.  If  the  in- 
stantaneous current  be  passed  through  the  movable 
coils  at  recurring  intervals  of  the  same  frequency  as 
the  currents  under  test,  the  movable  coils  will  all  take 
permanent  deflections  which  are  proportional  to  the 
corresponding  instantaneous  values  of  the  alternating 
currents.  By  changing  the  instant  of  contact  at  the 
contact  maker,  the  point  at  which  the  instantaneous 
current  passes  through  the  movable  coils  may  be  made 
coincident  with  any  point  on  the  alternating  current 
waves.  Thus  various  points  on  the  waves  may  be 
simultaneously  determined,  and  the  curves  may  be 
plotted. 

The  electrodynamometers  must  be  calibrated,  but 
this  may  be  readily  accomplished  by  passing  known  con- 
tinuous currents  through  the  fixed  coils  of  the  instru- 
ments while  the  regular  interrupted  test  current  is 
passed  through  the  movable  coils.  A  calibration  curve 
may  be  plotted  from  these 
observations.  To  assure 
the  constancy  of  the  inter- 
rupted test  current  during 
a  series  of  observations,  a 
d'Arsonval  galvanometer 
may  be  inserted  in  the  cir- 
cuit. In  Dr.  Duncan's 
work  it  was  found  neces- 
sary to  make  the  resist- 


Fig.  133 


ance  of  the  circuit  of  the  movable  coils  quite  large  (1000 
ohms)  in  order  to  eliminate  the  effect  of  the  variable 
contact  resistance  at  the  contact  maker.  In  order  that 


300  ALTERNATING   CURRENTS. 

the  current  through  the  coils  should  be  brief  and  well 
denned  a  condenser  discharge  was  found  advantageous 
(Fig.  133)-* 

7.  Bedell's  Method  (1893).  Dr.  Frederick  Bedell  of 
Cornell  University,  with  others,  has  lately  made  a  dis- 
position of  the  instruments  which  is  advantageous  in 
many  cases.  Each  of  the  methods  thus  far  described 
depends  upon  the  use  of  a  special  instrument  or  of 
instruments  that  are  difficult  to  handle  satisfactorily. 
On  the  other  hand,  electrostatic  voltmeters,  which 
might  be  made  to  replace  the  usual  instruments  and  are 
portable,  generally  have  a  scale  which  may  be  read  over 
only  a  limited  range.  An  instrument  reading  up  to 

J5O  volts,  for  instance, 
is  likely  to  give  very 
poor  indications  below 
60  volts.  In  order  that 
such  an  instrument  may 
be  used,  Dr.  Bedell  ar- 

Fig.  134 

ranes    it    with    a   con- 


CONTACT-MAKER 


l 

VOLT-^N 
METER \) 

[K   1 


denser  as  in  Fig.  134.  This  condenser  is  kept  charged 
to  a  known  potential  which  is  sufficient  to  bring  the 
needle  of  the  voltmeter  to  a  satisfactory  position  on  the 
scale.  That  is  to  say,  the  condenser  serves  to  displace 
the  zero  of  the  voltmeter  scale  a  known  amount.  The  val- 
ues of  instantaneous  pressures  read  on  the  voltmeter  are 
then  equal  to  the  indications  minus  the  initial  readings.! 
8.  Pupiris  Resonance  Analysis  (1893).  Dr.  M.  I. 
Pupin  of  Columbia  College  has  devised  and  experi- 

*  Trans.  Amer.  Inst.  E.  E.,  Vol.  9,  p.  179. 
t  Ibid.,  Vol.  10,  p.  503. 


CHARACTERISTICS,   REGULATION,   ETC.         301 


CONTACT  PIN 


Fig.  135 


mented  with  a  method  for  determining  by  resonance 
the  various  harmonics  which  enter  into  alternating-cur- 
rent curves.  If  the  sinusoidal  harmonics  are  fully 
known,  the  principal  curve  may  be  drawn  (Sect.  30).* 

Professor  Ayrton  several  years  ago  proposed  a  plan 
for  determining  the  sinusoidal  components  of  a  current 
curve  by  means  of  the  vibrations  of  a  stretched  wire. 

79.  Contact  Makers.  —  The  earliest  and  simplest  con- 
tact maker  was,  as  already  pointed  out,  simply  an  insu- 
lated pin  set  in  the  shaft 
of  the  alternator  furnish- 
ing the  current  for  the 
test.  With  this  was  a 
brush  so  arranged  as  to 
make  contact  with  the  pin 
at  any  desired  point  in 
the  revolution.  This  arrangement  is  often  inconvenient 
of  application,  and  is  likely  to  give  rather  irregular 
results.  The  contact  is  likely  to  be  variable  in  resist- 
ance and  as  the  brush  wears,  the  duration  of  contact 
varies.  Each  of  these  points  introduces  errors  of 
greater  or  less  magnitude,  depending  upon  the  condi- 
tions of  the  test.  Various  refinements  of  construction 
have  been  introduced  by  experimenters  in  order  that 
the  defects  of  the  contact  makers  may  be  eliminated. 
Figures  135  to  139  show  the  contact  makers  used  by  Jou- 
bert,  Searing  and  Hoffman,  Ryan,  Duncan,  and  Blondel.f 

*  Pupin,  Trans.  Amer.  Inst.  E.  E.,  Vol.  n,  p.  523. 

t  Comptes  Kendus,  Vol.  91,  p.  161;  Jour.  Franklin  Institute,  Vol.  128, 
p.  93;  Trans.  Amer.  Inst.  E.  E.,  Vol.  7,  p.  3;  Ibid.,  Vol.  9,  p.  181; 
La  Lumfcre  Electrique,  Vol.  41,  p.  512. 


302 


ALTERNATING  CURRENTS. 


The  contact  makers  used  by  each  of  these  experi- 
menters depend  upon  the  mechanical  contact  between 
a  point  and  a  brush  or  spring,  and  therefore  do  not 
entirely  avoid  the  difficulties  from  poor  or  variable  con- 
tacts. If  a  contact  of  absolute  uniformity  were  assured, 
special  instruments  would  not  be  necessary  for  taking 
the  indications  in  determining  pressure  and  current 


Fig.  136 

curves,  because  the  indications  of  a  sensitive  electro- 
dynamometer  might  then  be  directly  used.  Professor 
Ryan  and  Dr.  Bedell  have  lately  made  an  ingenious 
arrangement  by  which  the  duration  and  resistance  of 
the  contact  are  made  quite  uniform.  The  arrangement 
is  shown  in  Fig.  140.  It  consists  essentially  of  a 
revolving  disc,  D,  attached  to  the  dynamo  shaft,  and  a 
stationary  graduated  head,  H.  From  the  revolving  disc 


CHARACTERISTICS,   REGULATION,  ETC.         303 


a  needle,  N,  projects.     To  this  one  dynamo  terminal  is 
attached.     Upon  the  graduated  head  an  insulated  brass 


Fig-.  137 

nozzle,  T,  is  mounted.     The  nozzle  has  a  fine  hole  in 
it,  and  is  so  mounted  that  a  thin  jet  of  water  flowing 


Fig-.  138 


from  it  is  cut  once  in  a  revolution  by  the  needle.     A 
connection  from  the  nozzle  to  the  indicating  instrument 


304 


ALTERNATING  CURRENTS. 


completes  the  contact  maker.  By  means  of  the  gradu- 
ated head  the  contact  may  be  made  at  any  desired  point 
of  the  revolution.  It  is  found  that  the  jet  may  be 
satisfactorily  maintained  from  a  jar  of  water  a  few  feet 
above  the  contact  maker.  The  nozzle  is  radial,  the  jet 
keeps  its  direction  for  some  little  distance  before  being 
broken  up,  and  the  needle  cuts  the  jet  quite  near  to  the 


Fig.  139 


Fig.  140 


nozzle  where  it  is  fairly  stiff.  Water  with  a  little  salt  in 
it  is  used,  as  pure  water  has  too  high  a  resistance,  and 
acidulated  water  corrodes  the  apparatus.* 

The  contact  makers  described  thus  far  have  been 
arranged  for  a  single  contact,  but  it  is  frequently  desira- 
ble to  make  simultaneous  observations  of  several  curves. 

*  Trans.  Amer.  Inst.  E,  E.,  Vol.  10,  p.  500. 


CHARACTERISTICS,    REGULATION,   ETC.         305 

When  Duncan's  method  is  not  available,  this  may  be 
readily  accomplished  by  using  a  contact  maker  with 
the  appropriate  number  of  contact  discs  on  the  same 
spindle  (Fig.  139).*  Then  a  satisfactory  instrument, 
such  as  an  electrostatic  voltmeter,  may  be  used  in  each 
circuit.  Sometimes  it  is  not  convenient  to  have  the 
contact  maker  attached  to  the  dynamo  shaft,  in  which 
case  it  may  be  attached  to  a  short  length  of  flexible 
shaft  (Fig.  141),  which  may  in  turn  be  attached  to  the 
dynamo  shaft.  When  connection  to  the  alternator  can- 


Fig.  141 

not  be  conveniently  made,  the  contact  maker  may  be 
driven  by  a  synchronous  motor  as  has  been  done  by 
Blondel,t  Siemens  and  Halske,  and  Fleming.^ 

Any  of  the  methods  in  which  a  reflecting  instrument 
is  used  may  be  made  continuously  self-recording  by  a 
proper  disposition  of  the  apparatus.  In  this  case  a 
beam  of  light  is  thrown  upon  the  mirror,  and  its  devi- 
ation is  recorded  by  means  of  a  moving  photographic 

*  See  Blondel,  La  Lumiere  Electrique,  Vol.  41,  p.  512. 
t  La  Lumiere  Electrique,  Vol.  50,  p.  476. 
\  London  Electrician,  Vol.  34,  p.  460. 

X 


306         ALTERNATING  CURRENTS. 

film.  In  order  that  the  complete  curve  may  be  thus 
recorded,  the  contact  points  must  be  caused  to  rotate 
continuously  around  the  spindle  of  the  contact  maker. 
A  form  of  contact  maker  designed  for  this  purpose  is 
shown  in  Fig.  139.  Since  the  needle  of  the  galvanom- 
eter or  electrometer  which  is  used  with  the  contact 
maker  must  rigidly  follow  the  intensity  of  the  current 
impulses,  the  instrument  must  be  truly  deadbeat  and 
have  little  inertia.  The  vibrations  of  a  telephone  dia- 
phragm have  been  used  to  replace  the  deviations  of 
a  galvanometer  or  electrometer  needle.* 

80.  Areas  of  Successive  Curves.  —  In  general,  obser- 
vations which  cover  one  complete  period  entirely  define 
the  curves  of  current  and  pressure.  Since  there  is  no 
continuous  transference  of  electricity  in  one  direction, 
the  areas  of  successive  loops  of  the  curves  should  be 
equal.  In  the  pressure  curves  produced  by  an  alter- 
nator, for  instance,  e  =  — >  and  N=  I  edt,  where  N  is 
at  J 

the  total  number  of  lines  of  force  passing  into  the 
armature  core  and  \  dt  is  the  length  of  the  period.  If 
N  and  T  are  constant,  as  would  be  the  case  for  an  alter- 
nator with  fixed  field  magnetism  and  a  rigid  armature 
shaft  which  is  driven  at  a  uniform  speed,  the  values  of 
the  successive  areas  must  be  equal.  On  account  of 
various  irregularities  in  the  construction  and  working 
of  alternators,  experimentally  determined  curves  are  not 
always  uniform.  In  fairly  large  commercial  machines 
the  differences  are  usually  not  greater  than  might  be 

*  See  Blondel,  La  Lumiere  Alectrique,  Vol.  41,  p.  401 ;   Froelich,  Elek- 
trotechnische  Zeitschrift,  Vol.  10,  p.  345. 


CHARACTERISTICS,   REGULATION,   ETC.          307 


s  s 


308  ALTERNATING   CURRENTS. 

caused  by  the  errors  of  observation  due  to  the  experi- 
mental determination,  and  appreciable  differences  in 
the  areas  of  successive  loops  of  the  curves  produced 
by  mechanically  rigid  machines  driven  at  a  uniform 
angular  velocity  are  not  to  be  expected,  except  pos- 
sibly when  the  machines  have  armatures  with  their 
halves  connected  in  parallel,  and  then  only  when  the 
magnetic  circuits  lack  symmetry  to  a  considerable 
degree.  In  the  case  of  certain  small  eight-pole  alter- 
nators, Dr.  Bedell  found  differences  in  the  areas  of  the 
consecutive  loops  which  are  scarcely  explainable  upon 
the  ground  of  errors  of  observation  or  of  variable 
speed.*  The  curves  given  by  two  of  these  machines 
in  one  complete  revolution  (four  complete  periods)  are 
shown  in  Fig.  142.  The  individual  areas  of  the  loops 
are  marked  upon  the  figure.  While  these  differ  as 
much  as  25  per  cent  amongst  themselves,  the  sums  of 
the  positive  and  negative  areas  differ  by  no  more  than 
might  be  caused  by  experimental  errors.  This  appar- 
ently shows  that  irregularities  in  the  magnetic  circuits 
and  in  the  armature  windings  may  in  some  cases  cause 
differences  in  the  successive  loops  of  the  curve  devel- 
oped in  one  revolution,  but  the  algebraic  summation  of 
the  areas  due  to  each  revolution  is  zero.  The  latter 
must  be  true,  or  there  would  be  a  continuous  flow  of 
electricity  in  one  direction.  The  fact  that  the  machines 
tested  by  Dr.  Bedell  had  notable  structural  weaknesses, 
leads  to  the  probability  that  the  springing  of  the  shaft 
or  other  parts  of  the  machine  may  have  caused  the 
unusual  result  which  he  found. 

*  Physical  Review,  Vol.  I,  p.  218. 


CHARACTERISTICS,   REGULATION,   ETC.         309 

81.  Determination  of  the  Effective  Values  of  Current 
or  Pressure  from  their  Curves.  —  It  is  often  desirable  to 
determine  effective  values  of  current  or  pressure  from 
the  experimentally  determined  curves.  In  this  case,  a 
second  curve  may  be  plotted,  the  ordinates  of  which  are 
equal  to  the  square  of  the  respective  ordinates  of  the- 
primary  curve.  The  square  root  of  the  mean  ordinate 
of  the  second  curve  is  the  effective  value  of  the  ordinates 
of  the  primary  curve.  The  mean  ordinate  of  any  curve 


Fig.  143 

is  readily  determined  by  measuring  its  area  by  plani- 
meter  and  dividing  the  area  by  the  length  of  the  base. 
The  effective  value  may  be  directly  derived  from  the 
primary  curve,  as  originally  shown  by  Steinmetz,*  if  it  is 
plotted  on  polar  coordinates,  taking  360°  to  a  complete 
period.  This  gives  a  symmetrical  curve  which  crosses 
the  origin  at  o°,  180°,  360°,  etc.  For  an  exact  sinusoid 
the  curve  is  of  the  form  shown  in  Fig.  143,  and  has  its 
maximum  value  positive  and  negative,  at  90°  and  270°. 

*  Trans.  Amer.  hist.  E.  £.,  Vol.    IO,  p.  527;   Elektrotechnische  Zeit- 
schrift,  June  20,  1890. 


3IO  ALTERNATING   CURRENTS. 

Each  loop  is  a  circle  with  the  pole  on  its  circumference 
and  the  initial  line  tangent  to  the  circumference,  the 
maximum  ordinate,  a,  being  equal  to  the  diameter.  The 
area  of  the  curve  in  this  form  may  be  shown  to  be  di- 
rectly proportional  to  the  effective  value  of  the  ordinates 
as  follows  :  In  the  case  of  a  sinusoidal  curve,  the  polar 
curve  has  the  equation  e  =  a  sin  a,  where  e  is  the  instan- 
taneous pressure  corresponding  to  an  angular  advance  a. 
In  plotting  the  curve,  values  of  e  are  laid  off  on  the 
radius  vectors  having  vectorial  angles  equal  to  the  cor- 
responding values  of  a,  and  a  line  is  drawn  through  the 
points  thus  located  (Fig.  143).  Each  loop  of  this  curve, 
that  is,  the  part  of  the  curve  taken  between  a  =  o°  and 
a=  1  80°,  or  a  —  180°  and  a=  360°  is  a  circle,  and  its 
area  is  A  —  \  W2,  where  d  is  the  diameter  of  the  circle. 
By  the  construction,  d  is  equal  to  a  of  the  formula 
e  =  a  sin  a,  and  the  area  of  a  loop  of  the  curve  is  there- 
fore A=\  ira*.  The  effective  ordinate  of  a  sinusoid  has 
already  (Vol.  I.,  p.  83)  been  shown  to  be 

77—  _!_  a 

-£  TT  ^roaz  —  ,-' 

V2  V2 


Consequently          E  =  \i.  =  798  VA 


7T 


This  may  be  taken  for  most  purposes  as  E=  . 
In  the  case  of  any  single-valued  function 

e  =  a  sin  a  -f-  b  sin  2  a  +  c  sin  3  a  -f-  etc. 

+  a'  cos  a  +  b'  cos  2  a  +  c1  cos  3  a  -f  etc., 

and  the  area  of  one  loop  of  the  polar  curve  representing 


312  ALTERNATING   CURRENTS. 

/*7T 

this  is  A  =  I    e^da.     The    mean   of   the  squared  ordi- 
nates  of  the  function  is  av.  e2=  -  1,   e*da.     The  effective 

TJ-C/" 

ordinate  is 


As  before,  this  may  be  taken  as  E  —  .8 ^J A. 

Figure  144  shows  the  curve  of  squared  ordinates  and 
the  polar  curve  for  the  pressure  wave  of  the  Stanley 
alternator,  to  which  reference  has  already  been  made. 


REGULATION   AND   COMBINED   OUTPUT.        313 


CHAPTER   VII. 

REGULATION  AND  COMBINED  OUTPUT. 

82.  Regulation  for  Constant  Pressure.  —  A.  Separately 
Excited  Alternator.  A  separately  excited  alternator  has, 
as  already  intimated,  no  inherent  tendency  towards 
regulation.  The  regulation  is  usually  effected  by  hand, 
either  by  means  of  a  hand  regulator  in  the  field  circuit 
of  the  shunt-wound  exciter  or  a  hand  regulator  directly 
in  series  with  the  alternator  fields.  The  adjustment  of 
these  regulators  may  be  performed  through  devices 
actuated  by  a  relay  placed  as  a  shunt  to  the  main  cir- 
cuit, but  this  is  considered  inadvisable  in  this  country 
and  the  use  of  automatic  regulators  with  alternators  is 
entirely  unknown ;  but  in  Great  Britain  and  Europe 
automatic  devices  are  used  in  many  large  plants. 

An  ingenious  device  which  seems  to  give  satisfaction 
is  made  by  the  firm  of  Ganz  &  Company  of  Buda  Pesth. 
The  essential  parts  of  this  regulator  are  a  solenoid 
which  is  connected  as  a  shunt  to  the  main  circuit.  This 
solenoid  attracts  an  iron  core  which  carries  a  mercury 
cup  at  its  top.  .  Since  the  current  which  circulates  in  the 
solenoid  depends  upon  the  pressure  of  the  main  circuit, 
the  position  of  the  core  with  its  mercury  cup  depends 
upon  the  pressure.  A  series  of  wires  of  graduated 
lengths  dip  into  the  mercury  cup  in  such  a  way  that 


ALTERNATING   CURRENTS. 

the  ends  of  more  or  less  of  them  are  immersed  as  the 
pressure  falls  and  rises  (Fig.  145).  The  wires  are  at- 
tached to  resistances  which  are  connected  in  the  field 
circuit  of  the  exciting  dynamo,  but  which  are  short-cir- 
cuited when  the  ends  of  the  wires  dip  into  th(.  mercury. 
It  is  desirable  to  keep  the  pressure  constant  at  the  point 


Fig.  145 


Fig.  146 


of  consumption  rather  than  at  the  dynamos,  and  Ganz  & 
Company  have  succeeded  in  arranging  their  regulators 
to  do  this  without  the  inconvenience  of  "pressure  wires" 
(i.e.  wires  which  run  from  the  centre  of  consumption  to 
the  generating  station  for  the  purpose  of  indicating  the 
pressure  of  consumption).  This  requires  that  the  press- 
ure acting  in  the  circuit  of  the  automatic  regulator  shall 
be  caused  to  remain  constant  as  the  dynamo  current 


REGULATION   AND   COMBINED   OUTPUT.        315 

increases,  while  at  the  same  time  the  dynamo  pressure 
increases  by  a  sufficient  amount  to  compensate  for  the 
fall  of  pressure  in  the  feeders  which  run  to  the  centre 
of  distribution.  In  other  words,  E  —  CR  must  be  kept 
constant,  E  being  the  dynamo  pressure,  C  the  current, 
and  R  the  resistance  of  the  feeders.  This  is  effected  as 
follows  (Fig.  146)  :  The  regulator  is  connected  to  the 
secondary  of  a  special  transformer  T,  which  is  con- 
nected in  parallel  across  the  feeders.  The  pressure  of 
the  secondary  of  this  transformer  is  proportional  to 
the  dynamo  pressure  E.  Another  transformer,  Tf, 
is  connected  with  its  primary  in  series  with  the  feed- 
ers. The  pressure  developed  in  the  secondary  of  this 
transformer  can  be  adjusted  so  as  to  be  practically 
equal  to  CR  for  all  values  of  the  current.  The  sec- 
ondary of  this  is  connected  in  series  with  the  secondary 
of  the  first  transformer,  and  in  such  a  way  that  their 
pressures  are  in  opposition.  Hence  a  voltmeter,  V,  con- 
nected across  the  terminals  of  the  two  secondaries  indi- 
cates a  pressure  which  is  proportional  to  the  pressure 
at  the  terminals  of  the  feeder,  or  E  —  CR.  If  the 
automatic  regulator  is  also  connected  across  the  termi- 
nals of  the  two  secondaries,  it  will  adjust  the  excitation 
of  the  alternator  so  that  E  —  CR  is  kept  constant  regard- 
less of  the  value  of  C.  In  the  figure,  5  is  the  solenoid 
of  the  regulator,  Rv  is  the  resistance  automatically  con- 
trolled by  the  solenoid  to  vary  the  excitation  of  the 
alternator,  and  Rlt  R2,  R3  are  resistances  in  circuit  with 
the  regulator  which  are  used  for  adjusting  it  to  give 
proper  indications  for  various  values  of  £7  and  R.* 

*  Fleming's  Alternate  Current  Transformer,  Vol.  II.,  p.  137. 


ALTERNATING   CURRENTS. 


The  device  here  used  for  obtaining  at  the  terminals 
of  the  regulator  a  pressure  which  is  proportional  to 
E  —  CR,  is  exactly  similar  in  operation  to  the  Westing- 
house,  so-called,  " compensated  voltmeter"  which  is  used 
in  this  country.  In  this  case  hand .  regulation  is  exclu- 
sively adopted,  but  it  is  desirable  to  give  a  constant 
pressure  at  the  point  of  consumption. 
To  avoid  the  expense  and  annoyance 
of  "pressure  wires"  the  station  volt- 
meter is  connected  in  series  with  the 
secondaries  of  a  parallel  and  a  series 
transformer  which  act  in  opposition 
(Fig.  147).  The  transformers  being 


AA/WW 


0 

0 

rrr        (]) 

•  15 
•  11 

'•••     •'•'#-':'>'••'-' 

•  13 

3« 

•  12 
•  11 

4*  * 

*10 

5       •     •     •     * 

. 

0                     '78 

0 

HAND   REGULATOR 


Fig-.  147 


properly  adjusted,  the  voltmeter  shows  at  all  times  the 
lamp  pressure  at  the  centre  of  consumption.*  In  the 
Westinghouse  apparatus,  several  terminals  are  brought 
out  from  the  secondary  of  the  series  transformer  to  a 


*  Compare  Fleming's  Alternate  Current  Transformer,  Vol.  II.,  p.  185- 


REGULATION   AND   COMBINED   OUTPUT.        317 


switch  like  that  shown  in  Fig.  147,  and  the  adjustment 
of  the  apparatus  to  compensate  for  any  line  drop  is  made 
by  changing  the  secondary  connections  and  so  chang- 
ing the  number  of  effective  secondary  turns  in  the 
regulator  circuit.  In  some  cases,  the  secondary  of  the 
series  transformer  is  not  connected  into  the  circuit  of 
the  regular  voltmeter  coil,  but  is  connected  to  an  aux- 
iliary coil  which  is  wound 
alongside  of  or  over  the 
main  coil. 

The  Westinghouse  Com- 
pany also  manufacture  a 
feeder  regulator  for  use  in 
plants  where  several  cir- 
cuits are  fed  from  one  alter- 
nator or  set  of  'bus  bars. 
This  is  essentially  a  special 
transformer  (Fig.  148)  with 
the  secondary,  CD,  con- 
nected in  series  with  one  

feed  wire,  and  the  primary, 

AB,  connected  across  the   ' 

mains;  the  pressure  in- 
duced in  CD  may  be  made  to  either  aid  or  oppose  the 
alternator  pressure  by  means  of  a  reversing  switch,  X. 
The  strength  of  the  induced  pressure  in  CD  may  be 
varied  by  changing  the  number  of  effective  turns  in 
either  CD  or  AB  by  means  of  movable  contacts. 
Figure  149  gives  a  view  of  the  transformer  with  the 
regulator  switches.  Similar  devices,  in  which  the  regu- 
lation is  effected  by  varying  the  position  of  the  primary 


BUS    BARS 


Fig-.  148 


ALTERNATING   CURRENTS. 


and  secondary  coils  with  respect  to  each  other,  or  of 
the  core  with  respect  to  both,  are  manufactured  by  the 
General  Electric  Company. 

In  an  English  plant,   for  which   the  machinery  was 
constructed  by  the  Electric  Construction   Corporation, 

another  plan  is  used 
in  regulating  separately 
excited  alternators.  In 
this  case  series-wound 
exciters  are  used,  the 
regulation  of  which  is 
effected  by  shunting 
their  fields.  The  shunt 
is  composed  of  a  liquid 
resistance  into  which 
dip  two  plates  which  are 
connected  across  the 
terminals  of  the  exciter 
fields.  These  plates  are 
raised  and  lowered  in  the 
liquid,  to  vary  the  re- 
sistance of  the  shunt, 
by  means  of  a  solenoid. 
This  in  turn  is  actuated 
by  an  ingenious  thermal 
relay,  which  consists  of 

two  stretched  wires  connected  to  the  secondary  of  a 
transformer.  The  primary  of  the  transformer  is  con- 
nected across  the  main  circuit  of  the  alternator  or 
across  the  terminals  of  one  coil  of  its  stationary  arma- 
ture. The  pressure  developed  in  the  transformer 


Fig-.  149 


REGULATION   AND    COMBINED   OUTPUT.        319 

secondary  is  therefore  proportional  to  the  alternator 
pressure.  When  the  latter  falls  below  normal,  less 
than  the  normal  current  flows  through  the  relay  wires, 
which  contract  enough  to  actuate  a  switch  which  causes 
the  solenoid  to  lift  the  electrolytic  plates  and  thus 
increase  the  resistance  across  the  fields  of  the  exciter. 
When  the  alternator  pressure  rises  above  the  normal, 
more  current  flows  through  the  relay  wires,  which,  by 
sagging,  actuate  the  regulating  apparatus  so  that  the 
plates  are  lowered  further  into  the  liquid.  In  this 
manner  the  fields  of  the  exciter  are  regulated  so  that 
the  alternator  pressure  is  kept  constant. 

In  this  country  self-regulation  of  alternators  is  pre- 
ferred to  automatic  regulation  by  external  devices.* 
This  is  effected  by  means  of  composite  windings  (Sect. 
71).  Composite-wound  alternators  may  be  best  treated 
as  separately  excited  alternators  with  a  certain  number 
of  self-excited  series  turns  on  the  field-magnets.  The 
self-excited  field  turns  are  usually  of  sufficient  number 
to  make  the  external  characteristic  a  nearly  straight 
horizontal,  or  slightly  rising  line.  The  predetermina- 
tion of  the  number  of  series  turns  required  to  give 
exact  compounding,  or  a  desired  degree  of  over-com- 
pounding, is  not  readily  accomplished  when  no  experi- 
mental data  of  the  machine  is  at  hand,  on  account  of 
the  complex  effect  of  self-induction  and  armature  reac- 
tions upon  the  pressure  of  the  machine  (compare  Sect. 
70).  The  compounding  that  gives  regulation  on  an  in- 
ductionless  load  evidently  may  fail  for  an  inductive 
load.  The  ratio  of  series  ampere-turns  per  pole  to 

*  Compare  Text-book,  Vol.  I.,  p.  216. 


320         ALTERNATING  CURRENTS. 

armature  ampere-turns  per  coil  which  is  required  to 
give  regulation  for  one  alternator  of  a  fixed  type,  will 
doubtless  give  equally  satisfactory  results  on  machines 
of  different  capacities  but  of  the  same  type ;  but  the 
marked  differences  in  the  magnitude  of  the  effects  of 
self-induction  and  armature  reactions  in  alternators  of 
different  types,  make  it  impossible  to  fix  any  ratio  that 
will  even  approximately  cover  all  types  of  machines. 
For  alternators  with  smooth-core  drum  armatures  the 
ratio  of  series  ampere-turns  per  pole  to  ampere-turns 
per  armature  coil  is  practically  unity.  In  machines 
with  ring,  pole,  and  toothed  armatures  the  ratio  is 
doubtless  considerably  greater.  In  machines  with  disc 
armatures  it  may  be  somewhat  smaller.  The  various 
arrangements  of  the  circuits  that  may  be  made  in  com- 
posite windings  have  already  been  pointed  out  (Sect. 
71).  It  is  quite  common  to  place  a  variable  shunt 
around  the  series  windings  so  that  the  magnetizing 
effect  may  be  varied,  exactly  as  is  done  in  compound- 
wound  continuous-current  dynamos  (Vol.  I.,  p.  225). 

83.  Regulation  for  Constant  Pressure. — B.  Self-ex- 
cited Alternator.  The  defect  in  self-regulation  of  an 
alternator  excited  from  an  independent  winding  on 
the  armature  is  practically  the  same  as  that  of  a  sepa- 
rately excited  machine.  In  a  shunt-wound  machine 
the  defect  is  greater,  as  already  stated  (Sect.  74). 
The  regulation  of  alternators  which  are  self-excited  in 
shunt  or  by  a  separate  exciting  coil,  can  only  be 
satisfactorily  effected  by  means  of  a  variable  resist- 
ance or  hand  regulator  placed  in  the  exciting  circuit. 
The  regulation  might  be  effected  by  moving  the  brushes 


REGULATION  AND   COMBINED   OUTPUT         321 


3,000- 


2,000- 


1,500- 


1,000- 


500- 


on  the  rectifying  commutator,  but  only  at  the  expense 
of  prohibitive  sparking  and  wear.  The  variable  rheostat 
may  be  operated  automatically  by  the  same  devices  that 
are  sometimes  used  with  separately 
excited  machines.  These  do  not  meet 
with  favor  in  America,  however. 

84.  Regulation  for  Constant  Current. 
—  Constant-current  alternators  may  be 
either  separately  or  self-excited.  Their 
regulation  is  made  entirely  inherent  by 
designing  their  armature  reactions  and  2,500- 
self-induction  to  be  so  great  that  the  -1JE 
current  cannot  rise  above  its  normal 
value.  The  armature  is  wound  to  gen- 
erate a  pressure  upon  open  circuit 
much  greater  than  that  required  for 
full  load,  and  hence  the  current  remains 
near  its  full  normal  value  up  to,  and 
even  considerably  beyond,  full  load. 
Such  machines  are  worked  on  short- 
circuit  without  injury,  but  if  the  circuit 
is  opened,  they  are  liable  to  injury  on 
account  of  the  excessive  open  circuit 
pressure  breaking  down  the  insulation. 
These  machines  are  really  worked  on 
a  part  of  the  characteristic  which  is 
caused  to  be  almost  vertical  on  account 
of  the  large  self-inductance  of  the  ar- 
mature. Constant-current  alternators  have  only  been 
used  for  arc-lighting.  The  external  characteristic  of  a 
Stanley  arc-light  alternator  is  given  in  Fig.  150. 


8         9       10 
Fig.  15O 


322  ALTERNATING   CURRENTS. 

85.  Connecting  Alternators  for  Combined   Output.  — 

The  conditions  required  for  successfully  connecting 
alternators  so  that  their  outputs  may  be  combined,  are 
quite  different  from  those  obtaining  in  the  case  of  con- 
tinuous-current machines.  In  order  that  the  output 
of  alternators  may  be  added,  it  is  evident  that  the  press- 
ure waves  impressed  by  them  upon  the  circuit  must  be  in 
exact  consonance.  That  is,  the  pressure  waves  must  be 
of  equal  period  or  in  Synchronism,  and  also  of  corre- 
sponding phase  or  In  Step  with  each  other.  If  this  is 
not  the  case,  the  machines  will  be  in  opposition  during 
all  or  a  portion  of  the  current  wave. 

86.  Alternators  in  Series. — The  alternators  will  be 
assumed  in  this  discussion  to  be  constructed  so  as  to 
give  equal  currents  at  a  fixed  frequency.     The  form  of 
the  current  waves  will  also  be  assumed  to  approximate 
a  sinusoid. 

In  Fig.  151  a  let  the  curves  A  and  A'  represent  the 
electric  pressure  waves  of  two  alternators  with  their  ar- 
matures connected  in  series,  the  machines  being  driven 
independently,  but  so  as  to  give  practically  the  same 
frequencies  and  pressures.  The  ordinates  of  curve  R 
are  the  algebraic  sums  of  the  corresponding  ordinates 
of  curves  A  and  A' ,  and  hence  curve  R  represents  the 
resultant  pressure  of  the  two  machines.  Curve  C  is 
assumed  to  be  the  curve  of  current  flowing  in  the  cir- 
cuit. Assuming  the  two  machines  to  be  running  syn- 
chronously, but  to  be  out  of  step  by  an  angle  2  6,  makes 
the  phase  difference  between  the  resultant  pressure  wave 
and  either  component  wave  ±  6.  Finally,  the  current 
lags  behind  the  resultant  pressure  by  an  angle  <f>  on 


REGULATION   AND    COMBINED    OUTPUT.        323 

account  of  self-inductance  in  the  circuit.  The  work  put 
into  the  circuit  by  either  machine  is  proportional  to  the 
algebraic  summation  of  the  products  of  the  ordinates  of 
the  respective  pressure  and  current  waves.  The  total 
work  done  in  the  circuit  is  equal  to  the  sum  of  the  prod- 
ucts of  the  ordinates  of  the  current  and  resultant  press- 
ure curves.  Therefore,  since  the  pressure  wave  of  the 


Fig-.  151  a 

lagging  machine  is  nearest  the  current  wave,  that  ma- 
chine furnishes  more  work  to  the  circuit  than  does  the 
leading  machine.  The  power  loops  for  the  two  machines 
are  shown  by  the  curves  a  and  a'  in  Fig.  151  b,  and  the 
power  delivered  to  the  circuit  by  the  two  machines  is  rep- 
resented by  the  heights  of  the  lines  xx  and  x1xt.  Were 
the  two  machines  rigidly  connected  together,  this  condi- 


324 


ALTERNATING   CURRENTS. 


tion  would  continue  indefinitely.  In  practice,  however, 
the  machines  are  driven  by  separate  belts  or  attached  to 
separate  engines,  and  the  lagging  machine,  being  heavily 
loaded,  tends  to  fall  further  behind  its  more  lightly 
loaded  mate,  and  a  still  greater  percentage  of  the  load 
is  thrown  upon  it.  At  the  same  time,  as  is  shown  by 
Fig.  1520:,  this  reduces  the  total  work  done  in  the  exter- 
nal circuit,  for  the  total  pressure  wave  is  now  of  less 
height  than  it  was  when  the  component  curves  were 
more  nearly  in  phase.  The  power  loops  for  the  condi- 


tion of  Fig.  152^  are  shown  in  Fig.  i$2b.  The  height 
of  the  line  xx  has  decreased,  and  that  of  x*x}  has  in- 
creased, but  the  sum  of  the  heights  is  less  than 
before.  The  tendency  of  the  lagging  machine  to  fall 
further  behind  continues  until  the  pressure  waves  of 
the  two  machines  are  exactly  opposed  (Fig.  153).  The 
machines  are  then  in  stable  equilibrium,  but  are  giv- 
ing no  energy  to  the  external  circuit.  If  the  ma- 
chines were  started  with  their  pressure  waves  in  exact 
step,  they  would  do  equal  work,  but  their  equilibrium 
would  be  unstable,  and  any  disturbance  of  their  rela- 


REGULATION   AND   COMBINED   OUTPUT.        325 


tions  would  cause  them  to  fall  into  opposition.     It  is 
therefore  not  possible  to  operate  alternators  in  series 

R 


Fig.  152  a 

on  an  inductive  circuit  unless  they  are  rigidly  united  by 
a  mechanical  coupling.*     This  result  also  follows  when 
a!  a' 


-x' 


— \  —  -x 


*  Compare   Hopkinson,  Some  Points  in  Electric  Lighting,  Proc.  Inst. 
C.  E.t  1883,  and  Theory  of  Alternating  Currents,  your.  Inst.  E.  E.,  Vol. 


326 


ALTERNATING   CURRENTS. 


the  normal  pressures  of  the  machines  are  different;  in 
which  case  the  pressure  impressed  on  the  circuit  when 
equilibrium  is  attained  is  the  difference  of  the  machine 
pressures.  If  there  were  no  inductance  or  capacity  in 
the  circuit  on  which  the  machines  were  working,  the 
resultant  pressure  and  current  would  have  the  same 
phase,  and  the  machines  would  be  in  equilibrium,  but 
the  equilibrium  would  be  unstable,  for  after  any  disturb- 
ance of  the  operation  of  the  machines  they  would  have 


Fig.  153 

no  tendency  to  return  to  their  former  operating  state. 
No  such  case  is  to  be  met  with  in  any  event,  because 
the  armature  windings  of  the  machines  introduce  self- 
induction  and  current  lag  into  the  circuit,  even  when 
the  external  circuit  is  non-inductive.* 

87.    Alternators    in    Parallel. — When    the    machines 
have  reached  opposition  of  phases,  as  explained  above, 

13,  1884,  p.  496;  Hopkinson's  Dynamo  Machinery,  p.  148;  Picou's  Ma- 
chines Dynamo- Electrique,  p.  279;  Kapp's  Dynamos,  Alternators,  and 
Transformers,  p.  420;  Thompson's  Dynamo- Electric  Machinery,  4th  eel., 
p.  689. 

*  The  condition  of  operation  of  two  alternators  connected  in  series 
on  an  inductive  circuit  is  perhaps  more  plainly  indicated  by  the  following 

'-. 


REGULATION   AND   COMBINED   OUTPUT.       327 

a  change  in  the  arrangement  of  the  circuit  puts  them 
at  once  in  parallel  and  in  step  for  working  in  the  cir- 
cuit; for,  it  will  be  seen  by  reference  to  Figs.  153  and 
154  that  when  machines  A  and  A'  are  in  opposition,  the 

figure  (Fig.  i),  than  by  the  curves  which  were  used  in  the  preceding 
demonstration,  and  which  follow  those  originally  presented  by  Dr.  John 
Hopkinson,  in  1883  (Proc.  Inst.  C.  E.,  1883  ;  Jour.  Inst.  E.  £.,  1884). 

Pressure  of  leading  machine  —  OA. 

Pressure  of  lagging  machine  =  OA' . 

Resultant  pressure  in  circuit  •=  OR. 

Current  in  circuit  =  OC. 

Power  given  to  circuit  by  first  machine  =  Oa  X  OC. 

Power  given  to  circuit  by  second  machine  =  Oa'  x  OC. 

Total  power  given  to  circuit  =  Or  xOC  =  (Oa  +  Oa'}  X  OC. 

It  is  evident  from  the  construction  that  as  the  angle  6  increases,  the 
length  of  OR  decreases,  and  also  that  Oa  decreases ;   but  Oa'  increases  for 


a  time  and  then  decreases  at  a  less  rate  than  Oa,  so  that  the  machines  tend 
to  get  farther  apart  in  phase.  When  6-  =  90°  —  0  the  length  of  Oa  van- 
ishes, the  first  machine  gives  no  power  to  the  circuit,  and  all  the  power  is 
furnished  by  the  second  machine.  When  6  approaches  more  nearly  90°, 
or  the  machines  are  approaching  opposition,  one  machine  may  actually 

•' 


328 


ALTERNATING  CURRENTS. 


points  R  and  5  must  at  every  instant  be  of  opposite 
sign,  and  that,  therefore,  the  machines  will  deliver  cur- 
rent through  the  circuit  m,  m,  m*  The  operation  of 
alternators  in  parallel  was  first  achieved  by  Wilde  in 


-  R 


Fig.  154 

i868,f  but  this  work  was  overlooked  during  the  period 
of  development  of  the  continuous-current  dynamo.  In 
1884  Dr.  John  Hopkinson  showed  by  mathematical 
analysis,  in  the  paper  already  referred  to,  the  impracti- 
cability of  working  alternators  in  series  and  the  prac- 

run  as  a  motor.  Of  course,  when  OR  decreases,  if  the  resistance  of  the 
circuit  is  unaltered,  the  current,  OC,  also  decreases,  but  the  relative  out- 
puts and  phases  of  the  machines  are  not  altered  thereby. 

In  case  there  is  a  capacity  in  the  circuit  which  is  sufficient  to  cause  the 
current  to  lead  the  resultant  pressure  by  an  angle  <t>,  the  condition  is  repre- 
sented by  Fig.  2.  In  this  case,  Oa  is  greater  than  Oa',  or  the  leading 
machine  furnishes  the  greatest  amount  of  power,  and  the  machines  tend  to 
come  together  and  run  in  series.  This  is  not  a  practical  condition,  how- 
ever, since  a  capacity  in  a  commercial  alternator  circuit  sufficient  to  give 
the  current  a  lead  is  practically  unknown. 

*  Compare  references  given  above,  and  Fleming's  Alternate  Current 
Transformer,  Vol.  II.,  p.  356. 

t  See  Philosophical  Magazine,  Vol.  37,  4th  series,  1869,  p.  54. 


REGULATION    AND   COMBINED   OUTPUT.        329 

ticability  of  working  them  in  parallel.  This  was  done 
without  a  knowledge  of  Wilde's  earlier  experiments, 
and  it  led  to  some  experiments  which  were  carried  out 
by  Hopkinson  and  Adams  upon  De  Meritens  magneto 
machines.*  These  experiments  fully  bore  out  Hopkin- 
son's  deductions,  but  their  practical  bearing  was  not 
fully  appreciated  until  a  few  years  later,  when  the  trans- 
former system  of  alternating-current  distribution  was 
developed.! 

88.  Synchronizers  and  Synchronizing.  —  Mechanical 
imperfections  in  engine  governors  and  machine  pulleys 
cause  slight  differences  in  the  speeds  of  machines  in- 
tended to  run  at  equal  velocities.  Consequently  it  is 
desirable  to  arrange  some  device  for  determining  the 
moment  a  machine  is  in  synchronism  with  one  with 
which  it  is  to  be  thrown  in  parallel.  When  the  alter- 
nators are  to  be  connected  in  parallel,  the  terminals  of 
each  may  be  connected  directly  to  appropriate  'bus  or 
main  conductors  through  convenient  indicating  instru- 
ments, switches,  and  safety  devices.  Before  switching 
a  new  machine  upon  the  'bus  conductors  it  must  be 
brought  to  normal  speed,  and  to  the  pressure  of  the 
other  machines.  Then  at  a  moment  when  it  is  in 
synchronism  and  in  step  with  the  pressure  wave  of 
the  'bus  bars,  it  may  be  switched  into  circuit  without 
causing  a  disturbance  among  the  other  alternators. 

Any  device  for  indicating  the  synchronous  relation  is 
called  a  Synchronizer  or  Phase  Indicator.  Its  simplest 

*  Adams,  Jour,  Inst.  E.  E.,  Vol.  13,  1884,  p.  515. 

t  Compare  Hopkinson,  Jour.  Inst.  E.  E.,  1884,  and  Dynamo  Machin- 
ery, p.  174;  Thompson's  Dynamo- Electric  Machinery,  4th  ed.,  p.  696,  etc. 


330 


ALTERNATING   CURRENTS.. 


form  for  low-pressure  machines  consists  of  one  or  more 
incandescent  lamps  in  series,  which  are  connected  as  in 
Fig.  155.  One  terminal  of  the  alternator  is  connected 
directly  to  a  'bus  conductor,  while  the  other  is  connected 
through  the  lamps  to  the  other  'bus.  When  the  press- 
ure waves  are  not  in  opposition,  the  lamps  will  be 
illuminated,  and  at  the  moment  of  opposition  the  illu- 
mination will  die  out.  If  the  frequencies  of  the  alter- 
nator and  the  circuit  differ  materially,  the  flashes  of 
illumination  or  "beats"  are  quite  rapid.  As  the  fre- 


\ 


Pig.  155 

quencies  approach  synchronism,  the  beats  lengthen  out, 
exactly  as  do  the  beats  of  two  tones  which  are  approach- 
ing unison.  The  alternator  should  be  connected  to  the 
'bus  bars  at  an  instant  of  no  illumination  during  a 
period  when  the  beats  are  fairly  long.  This  indicates 
that  the  pressure  of  the  alternator  is  in  synchronism 
with  that  of  the  circuit  and  that  it  is  in  proper  step 
or  phase.  Continued  illumination  or  darkness  of  the 
lamps  under  these  circumstances  can  only  occur  when 


REGULATION   AND   COMBINED    OUTPUT.        331 


the  machines  produce  the  same  pressure  and  run  at 
absolutely  the  same  frequencies,  which  is  not  a  practi- 
cal occurrence  unless  the  machines  are  rigidly  connected 
together. 

Since  alternators  are  commonly  built  for  high  press- 
ures, it  is  usual  to  use  a  transformer  with  the  synchronizer 
lamps.  The  primary  circuit  of  this  transformer  may 
be  composed  of  either  one  or  two  windings.  When 


Fig.  156 

it  is  composed  of  one  winding,  one  terminal  of  the 
alternator  is  connected  to  a  'bus  bar  through  it,  the 
other  terminal  of  the  alternator  being  connected  di- 
rectly to  the  other  'bus  (Fig.  156).  In  this  case  the 
lamp  on  the  secondary  circuit  acts  exactly  as  the  syn- 
chronizer lamps  already  described.  When  the  primary 
is  composed  of  two  circuits,  one  is  connected  between 
the  'bus  conductors  and  the  other  between  the  termi- 
nals of  the  alternator  (Fig.  157).  In  this  case  the 
proper  phase  relation  for  switching  the  alternator  into 


ALTERNATING    CURRENTS. 


circuit  may  be  indicated  either  by  darkness  or  by  illu- 
mination of  the  lamps,  depending  upon  the  connection 
of  the  synchronizer  primaries.  This  arrangement  is 
advantageous,  since  it  allows  the  use  of  a  double-pole 
switch  at  the  dynamo,  while  the  previously  described 


I      / 


arrangements  require  the  use  of  single-pole  switches. 
The  latter  arrangement  may  be  modified  by  using  two 
separate  transformers,  the  primary  of  one  being  con- 
nected to  the  alternator  and  that  of  the  other  to  the 
circuit.  The  secondaries  are  connected  together  in 
series  with  one  or  two  lamps  (Figs.  158  and  159).  If 
the  secondaries  are  connected  directly  in  series,  as  in 


REGULATION   AND    COMBINED   OUTPUT.        333 


Fig.  158,  darkness  of  the  lamps  indicates  the  instant 
for  connecting  the  alternator  to  the  'bus  conductors. 
If  the  secondaries  are  cross-connected,  as  in  Fig.  159, 
maximum  illumination  of  the  lamps  indicates  the  mo- 
ment when  the  machines  are  in  proper  step.  In  this 
country  the  general  practice  has  been  to  connect  the 


A2 


A/WVWWWWWVV 


Fig.  158 

synchronizer  so  that  darkness  indicates  when  the  ma- 
chines are  in  step.  This  has  the  evident  advantage 
that  darkness  is  a  condition  which  is  more  readily  dis- 
tinguished than  the  condition  of  maximum  brightness. 
This  practice,  however,  does  not  seem  to  be  always 
followed  in  England  and  Europe.* 

In  any  of  these  methods,  the  lamps  may  be  replaced 

*  Compare  Fleming's  Alternate  Current  Transformer ;  Vol.  II.,  pp.  163 
and  362;  Kapp's  Alternating  Currents  of  Electricity,  p.  129;  Gerard's 
Lemons  sur  /' ' £lectricite,  3d  ed.,  Vol.  I,  p.  557. 


334 


ALTERNATING  CURRENTS. 


by  a  sensitive  high   resistance  alternating-current  am- 
peremeter or  galvanometer  which  is  dead-beat. 

An  ingenious  device  for  use  as  a  synchronizer  has 
lately  been  developed  by  the  General  Electric  Company. 
This  consists  of  two  electromagnets  made  with  iron 
wire  cores.  The  windings  of  these  are  connected 
respectively  to  the  alternator  and  the  circuit.  Each 
magnet  has  placed  in  front  of  it  an  iron  diaphragm 
which  emits  a  tone  which  has  a  pitch  due  to  the 


A2 


Fig-.  159 


frequency  of  the  current  flowing  in  the  winding.  In 
front  of  the  magnets  are  placed  resonators  which  mag- 
nify the  sound  emitted  by  the  diaphragms.  When  the 
two  tones  are  not  in  exact  harmony,  interference 
causes  beats,  and  the  synchronizer  emits  an  inter- 
mittent sound.  If  the  speed  of  the  machines  is 
brought  nearer  and  nearer  to  synchronism,  the  beats 


REGULATION   AND   COMBINED   OUTPUT          335 

become  less  rapid.  At  exact  synchronism,  the  beats 
die  out  and  a  clear  tone  results.  The  alternator  may  or 
may  not  be  in  step  when  the  clear  tone  is  given,  but  it 
may  be  safely  thrown  into  circuit,  for  the  interaction 
of  the  current  waves  will  bring  it  into  proper  phase 
relations.  This  synchronizer  was  designed  for  use  with 
synchronous  motors.  It  cannot  give  satisfaction  with 
alternators  that  are  furnishing  constant  pressure  current 
for  incandescent  lighting,  since  placing  an  alternator  in 
the  circuit  while  it  is  out  of  step  is  likely  to  momen- 
tarily disturb  the  pressure. 

It  is  entirely  possible  to  do  without  a  synchronizer 
when  connecting  alternators  in  parallel,  provided  they 
are  driven  at  approximately  equal  speeds.  In  this  case 
if  the  alternator  is  brought  to  the  proper  pressure  and 
is  then  connected  to  the  'bus  bars,  the  machine  reac- 
tions will  bring  it  into  step.  This  plan  was  practised 
to  some  extent  in  one  or  two  earlier  American  plants, 
but  it  is  vicious  in  its  working.  Throwing  machines 
onto  the  'bus  bars  under  these  conditions  usually  causes 
a  disturbance  in  the  pressure,  and  it  also  doubtless 
strains  the  armature  of  the  alternator  on  account  of  the 
sudden  torque  impressed  upon  it  to  bring  it  into  step. 

89.  Usual  Practice  with  Reference  to  Parallel  Opera- 
tion. —  Parallel  working  of  alternators  has  not  been  usual 
in  this  country  heretofore,  though  it  is  quite  a  common 
practice  in  Europe.  Several  of  the  earlier  American 
plants,  installed  by  the  Westinghouse  Electric  Company, 
were  arranged  for  parallel  working ;  but  the  plan  was 
quickly  abandoned  on  account  of  the  machines  dividing 
their  load  unevenly  and  thus  causing  trouble.  This 


336         ALTERNATING  CURRENTS. 

difficulty  was  quite  similar  in  many  respects  to  that 
encountered  in  the  early  endeavors  to  operate  compound 
continuous-current  dynamos  in  parallel.  It  was  usual 
in  these  early  alternating  plants,  as  it  is  now,  to  belt  the 
alternators  to  independent  engines.  On  account  of  the 
unsatisfactory  results  met  in  the  early  attempts  at  par- 
allel working,  it  has  come  to  be  the  almost  universal 
practice  in  this  country  to  operate  alternators  on  sepa- 
rate circuits.  This  introduces  some  complications  in 
the  station  switch-board  arrangements,  on  account  of 
the  necessity  of  making  the  connections  of  machines 
and  circuits  so  flexible  that  they  may  be  intercon- 
nected in  any  manner ;  but  satisfactory  switching  ar- 
rangements may  be  very  satisfactorily  accomplished. 
A  common  arrangement  of  switch  boards  for  this  pur- 
pose is  shown  in  Fig.  160,  where  double-pole  throw- 
over  switches  are  connected  in  such  a  manner  that  any 
feeder  may  be  connected  to  any  alternator  in  the  sys- 
tem. In  some  of  the  later  arrangements  for  large 
stations,  plugs  and  cords  are  arranged  to  be  used  in 
combination  with  the  throw-over  switches  (Fig.  161). 
By  these  arrangements  the  feeders  may  be  almost 
instantly  transferred  from  one  alternator  to  another,  so 
that  any  readjustment  of  the  alternator  loads  may  be 
made  without  more  disturbance  than  a  mere  wink  of  the 
lamps.  It  is  evident,  however,  that,  with  this  arrange- 
ment, no  feeder  can  be  designed  to  supply  a  district 
demanding  a  greater  output  than  that  of  the  station 
power  unit.  Figure  160  shows  an  arrangement  for  two 
alternators  and  four  feeders,  and  Fig.  161  an  arrange- 
ment for  three  alternators  and  three  feeders. 


REGULATION   AND   COMBINED   OUTPUT.        337 


Fig.  160 


338 


ALTERNATING   CURRENTS. 


90.  Elements  which  affect  the  Success  of  Parallel 
Operation.  —  On  account  of  the  development  of  long- 
distance power  transmission  plants  of  great  magnitude, 
in  which  alternating  currents  are  used,  parallel  working 
of  alternators  seems  likely  to  soon  become  common  in 


Co— o)  Co— o)       (o— o)  Co— Q) 


o 


FEEDER 
REGULATOR 


FEEDER 
REGULATOR 


FEEDER 
REGULATOR 


Fig.  161 

this  country.  In  such  plants  parallel  working  is  condu- 
cive to  convenience  and  reliability  in  operation.  It  also 
permits  a  saving  in  the  cost  of  line  insulation,  where 
high  pressures  are  used,  by  allowing  a  concentration 
of  conductors.  Parallel  working  of  alternators  is  also 
advantageous  in  large  electric  light  stations,  where  it 


REGULATION   AND   COMBINED    OUTPUT.        339 

may  be  made  conducive  to  convenience,  reliability,  and 
economy  of  operation.  The  subject  is  therefore  one 
of  considerable  interest  to  us.  There  is  a  considera- 
ble disagreement  amongst  builders  of  alternators,  and 
others,  as  to  why  some  types  of  alternators  apparently 
operate  well  in  parallel,  while  other  types  do  not. 
There  is  also  an  apparent  disagreement  as  to  what  con- 
stitutes successful  parallel  operation.  This  is  a  pure 
question  of  practice,  and  we  must  therefore  rely  upon 
results  that  have  been  and  are  being  attained  in  central 
station  work.  Mr.  W.  M.  Mordey  seems  to  have  been 
the  first  to  clearly  point  the  way  to  truth  in  the  case,*  as 
he  was  the  first  to  point  out  categorically  some  of  the 
relations  between  the  continuous-current  dynamo  and 
motor,  j- 

The  three  elements  causing  the  greatest  friction  in 
the  discussion  of  this  subject  are  the  effects  of  fre- 
quency, of  the  form  of  the  pressure  curve,  and  of  self- 
induction.  And  it  is  well  in  such  a  discussion  to 
consider  with  particular  attention  these  points:  (i)  the 
effect  of  frequency  ;  (2)  the  effect  of  armature  induct- 
ance ;  (3)  the  effect  of  the  form  of  the  current  and 
pressure  curves  ;  (4)  the  effect  of  regulation  by  varying 
the  excitation  ;  (5)  uniformity  of  angular  velocity.  None 
of  these  points  have  been  so  thoroughly  investigated  by 
experiment  as  to  be  decided  with  entire  conclusiveness, 
and  the  mathematical  investigations  have  in  many  cases 
been  based  upon  erroneous  premises,  and  are  therefore 


*  Alternate  Current  Working,  Jour.  Inst.  E.  £.,  Vol.  18,  p.  583. 
t  Philosophical  Magazine,  Vol.  21,   5th  series,    1886,  p.   20,  and    The 
London  Electrician,  Vol.  1 6,  p.  193. 


340  ALTERNATING   CURRENTS. 

incorrect;  but  the  experimental  investigations  have  been 
sufficiently  extended  to  give  a  satisfactory  clue  to  the 
true  conclusions. 

Successful  parallel  working  of  alternators  requires  : 
(i)  that  they  synchronize  readily  when  driven  at  speeds 
which  differ  by  a  few  per  cent  ;  (2)  that  they  shall  in- 
stantly fall  into  step  if  thrown  into  parallel  when  they  are 
practically  synchronized,  but  are  out  of  step  ;  (3).  that 
they  shall  continue  to  operate  in  parallel,  dividing  all 
loads  proportionally  under  the  varying  conditions  of  ser- 
vice and  attention ;  and  (4)  that  the  wattless  or  synchro- 
nizing current  passing  between  the  machines,  but  not  into 
the  external  circuit,  shall  be  small.  The  latter  condition 
requires  that  the  apparent  watts  passing  into  the  exter- 
nal circuit  shall  be  appreciably  equal  to  the  sum  of  the 
apparent  watts  delivered  by  the  individual  machines. 
If  the  machines  do  not  have  a  strong  inherent  tendency 
to  remain  in  step,  they  are  likely  to  "  seesaw "  or 
"  hunt "  ;  that  is,  one  machine  takes  the  lead,  then  falls 
behind,  and  after  a  time  again  takes  the  lead,  repeating 
this  operation  continually.  When  one  of  the  machines 
is  leading  or  lagging  with  respect  to  its  mates,  it 
develops  during  alternate  portions  of  the  periods  a 
higher  and  a  lower  pressure  than  its  mates.  It  is  there- 
fore alternately  electrically  driving  and  being  driven  by 
its  mates.  This  results  in  a  considerable  flow  of  watt- 
less current  which  interferes  with  regulation,  overloads 
the  machines  beyond  the  demands  of  the  external  cir- 
cuit, and  causes  irregular  and  unsatisfactory  working. 
It  is  not  sufficient  proof  of  the  adaptability  of  alternat- 
ors for  parallel  working  to  show  that  when  two  machines 


REGULATION   AND   COMBINED    OUTPUT.        341 

are  belted  to  separate  engines,  one  will  drive  the  other 
as  a  motor  if  the  steam  is  shut  off  one  of  the  engines. 
We  might  equally  say  the  proof  that  shunt-wound  con- 
tinuous-current dynamos  will  work  satisfactorily  in  par- 
allel is  made  when  it  is  shown  that  one  will  continue  to 
run  (as  a  motor)  when  the  steam  is  shut  off  its  engine. 

91.  The  Effect  of  Frequency  on  Parallel  Operation.  — 
The  parallel  operation  of  alternators  was  practised  in 
the  earlier  plants  installed  by  the  Westinghouse  Electric 
Company  in  this  country.  In  this  case  the  frequency 
'was  about  133.  The  machines  worked  together  quite 
well  when  carrying  full  load,  but  when  their  loads 
changed  they  would  not  properly  divide  the  load,  which 
led  to  hunting,  and  consequent  injury  to  the  service 
and  damage  to  the  machines.  At  the  best,  parallel 
working  increased  the  attention  required  at  the  alter- 
nators to  a  great  degree. 

Thomson-Houston  alternators  giving  a  frequency  of 
125  are  worked  in  parallel  with  apparent  satisfaction 
in  London,  England,  St.  Brieux,  France,  and  elsewhere. 

The  classical  experiments  of  Dr.  John  Hopkinson  in 
paralleling  alternators  were  performed  with  De  Meritens 
permanent-magnet  alternators,  giving  a  frequency  of 
about  1 20.  Of  the  results  obtained  in  these  experi- 
ments, Dr.  Hopkinson  says:  "The  two  machines  for 
Tino  were  driven  from  the  same  countershaft  by  link 
bands,  at  a  speed  of  850  to  900  revolutions  per  minute ; 
the  pulleys  on  the  countershaft  were  sensibly  equal  in 
diameter,  but  those  on  the  machines  differed  by  rather 
more  than  a  millimeter,  one  being  300,  the  other  299 
millimeters  in  diameter  ;  thus  the  machines  had  not, 


342         ALTERNATING  CURRENTS. 

when  unconnected,  exactly  the  same  speed.  The  pul- 
leys have  since  been  equalized.  The  bands  were  of 
course  put  on  as  slack  as  practicable,  but  no  special 
device  for  adjusting  the  tightness  of  the  bands  was 
used.  The  experiment  succeeded  perfectly  at  the  very 
first  attempt.  The  two  machines,  being  at  rest,  were 
coupled  in  series,  with  a  pilot  incandescent  lamp  across 
the  terminals  ;  the  two  bands  were  then  simultaneously 
thrown  on ;  for  some  seconds  the  machines  almost 
pulled  up  the  engine.  As  the  speed  began  to  increase, 
the  lamp  lit  up  intermittently,  but  in  a  few  seconds 
more  the  machines  dropped  into  step  together,  and  the 
pilot  lamp  lit  up  to  full  brightness  and  became  perfectly 
steady,  and  remained  so.  An  arc  lamp  was  then  intro- 
duced, and  a  perfectly  steady  current  of  over  200 
amperes  drawn  off  without  disturbing  the  harmony. 
The  arc  lamp  being  removed,  a  Siemens  electrodyna- 
mometer  was  introduced  between  the  machines,  and  it 
was  found  that  the  current  passing  was  only  18  am- 
peres ;  whereas,  if  the  machines  had  been  in  phase  to 
send  the  current  in  the  same  direction,  it  would  have 
been  more  than  ten  times  as  great.  On  throwing  off 
the  two  bands  simultaneously,  the  machines  continued 
to  run  by  their  own  momentum,  with  retarded  velocity. 
It  was  observed  that  the  current,  instead  of  diminishing 
from  diminished  electromotive  force,  steadily  increased 
to  about  50  amperes,  owing  to  the  diminished  electrical 
control  between  the  machines,  and  then  dropped  to 
zero  as  the  machines  stopped."  * 

*  Theory  of  Alternating   Currents,  Jour.  Inst.  E.   E.,  Vol.   13,   and 
Hopkinson's  Dynamo  Machinery  and  Allied  Stibjects,  p.  175. 


REGULATION   AND   COMBINED    OUTPUT.        343 

The  Mordey  alternator  with  a  disc  armature  is 
reported  to  operate  well  in  parallel  at  a  frequency  of 
100.  Mr.  Mordey  says  :  "With  regard  to  parallel  work- 
ing, I  can  only  say  that  we  find  nothing  in  practice  to 
lead  us  to  suppose  that  reducing  the  rate  (frequency) 
would  improve  the  working.  We  have  no  difficulty  in 
parallel  working  at  100  periods  per  second,  and  there- 
fore cannot  improve  in  this  respect."  The  Mordey 
alternator  is  working  in  parallel  in  a  number  of  English 
and  European  plants  with  apparent  satisfaction.  In 
experimenting  with  these  machines,*  Mr.  Mordey  made 
the  following  tests  : 

"(i)  The  alternators  were  run  up  to  full  speed,  and 
each  excited  to  give  2000  volts.  When  in  phase,  they 
were  switched  parallel  without  any  external  load,  and 
without  any  impedance  coils  or  resistance  between 
them.  They  ran  in  parallel  perfectly. 

"  (2)  A  considerable  inductionless  load  was  then  put 
on,  varied,  and  taken  off.  They  ran  equally  well  under 
all  circumstances. 

"  (3)  They  were  uncoupled,  and  then,  the  load  being 
connected  to  the  mains,  they  were  suddenly  and  simul- 
taneously switched  parallel  and  on  to  the  mains  with 
perfect  success. 

"  (4)  One  alternator  was  excited  to  give  1000  volts, 
the  other  giving  2000  volts.  They  were  then  switched 
parallel,  and  went  into  step  perfectly,  giving  a  terminal 
P.  D.  of  about  1500  volts.  No  impedance  or  resistance 
was  used  in  this  or  any  other  case.  A  load  was  then 
put  on  without  affecting  their  behavior. 

*  Alternate  Current  Working,  Jour.  Inst.  E.  £.,  Vol.  1 8. 


344         ALTERNATING  CURRENTS. 

"  (5)  With  one  machine  at  1000  volts,  and  the  other 
at  2000  volts,  they  were  switched  parallel  when  out  of 
phase,  and  instantly  went  into  step.  A  large  current 
appeared  to  pass  between  them  for  a  fraction  of  a  sec- 
ond, but  not  nearly  long  enough  to  enable  it  to  be 
measured  or  to  do  any  harm. 

"  (6)  They  were  then  left  running  parallel  while  one 
was  disconnected  from  the  engine  by  its  belt  being 
shifted  from  the  fast  to  the  loose  pulley.  It  continued 
to  run  as  a  motor  synchronously.  A  load  of  lamps  was 
at  the  same  time  on  the  circuit. 

"  (7)  The  two  machines  were  then  uncoupled,  and 
excited  up  to  2000  volts.  They  were  then  switched 
parallel  when  out  of  phase,  and  without  any  external 
load,  and  went  into  step  instantly. 

"  (8)  Whilst  running  as  in  (7),  steam  was  suddenly 
and  entirely  shut  off  one  engine.  The  alternators  kept 
in  step  perfectly,  one  acting  as  a  motor,  and  driving  the 
large  engine  and  all  the  heavy  countershafting  and 
belts.  It  was  impossible  to  tell,  except  by  the  top  of 
the  belt  becoming  tight  instead  of  the  bottom,  which 
machine  was  the  motor. 

"  To  find  the  power  exerted  by  the  alternator  acting 
as  a  motor  in  (8),  a  direct-current  motor  was  put  in  its 
place,  and  the  power  required  to  drive  the  engine  and 
shafting  was  found  to  be  20  horse  power." 

The  capacity  of  the  alternators  here  experimented 
with  was  about  50  horse  power. 

Siemens  alternators,  connected  directly  to  their  en- 
gines, are  operated  in  parallel  at  Bristol,  England.* 

*  London  Electrical  Review,  Vol.  34,  p.  274. 


REGULATION    AND    COMBINED   OUTPUT.        345 

Ferranti  alternators  with  disc  armatures,  giving  a 
frequency  of  83,  are  worked  together  in  England  and 
Europe.  At  the  Deptford  station,  in  London,  Ferranti 
alternators  of  two  sizes,  625  horse  power  and  1250  horse 
power,  are  worked  parallel,  though  their  normal  pressure 
is  different,  and  the  smaller  machines  are  therefore  con- 
nected up  to  the  circuit  through  a  transformer.* 

Elwell-Parker  alternators,  giving  a  frequency  of  80, 
are  reported  to  work  together  with  some  satisfaction. 
These  machines  are  somewhat  like  the  American  type 
turned  inside  out ;  that  is,  they  have  the  equivalent 
of  a  drum  armature,  which  surrounds  the  revolving 
field  magnets,  f 

In  certain  European  plants  Kapp  alternators  have 
operated  in  parallel.  These  machines  have  ring  arma- 
tures, and  give  a  frequency  of  70. 

Stanley  two-phase  alternators  with  frequencies  of  60 
and  125  are  running  in  parallel  with  perfect  success  in 
this  country. 

The  Gordon  alternators,  which  were  among  the  earli- 
est to  be  used  in  commercial  service,^  were  shown  to  be 
capable  of  operating  in  parallel.  Of  this,  however,  Mr. 
Gordon  said  :  "  We  know  that  experiments  have  been 
made  by  coupling  a  number  of  small  alternate-current 
machines  together,  and  at  the  South  Foreland  (Hopkin- 
son's  experiment)  they  were  successful,  but  that  was 
because  they  were  working  on  arc  lamps.  Many  of  us 
have  tried  them,  and  they  will,  on  trial,  work  together, 

*  Fleming's  Alternate  Current  Transformer,  Vol.  II.,  p.  359. 
t  Fleming's  Alternate  Current  Transformer,  Vol.  II.,  p.  222 ;    Mordey, 
Jour.  Inst.  E.  E.,  Vol.  18,  p.  588. 
|  See  Gordon's  Electric  Lighting. 


346         ALTERNATING  CURRENTS. 

no  doubt ;  but  they  do  not  work  together  till  they  have 
run  for  three  or  four  minutes  ;  they  will  in  that  time 
jump,  and  that  jumping  will  take  months  of  life  out  of 
the  40,000  lamps.  That  alone  is  rather  a  serious  diffi- 
culty in  coupling  machines  together,  and  I  think  we  may 
take  it  in  practice  —  I  am  not  speaking  about  the  labora- 
tory, or  experiments  —  we  do  not  couple  machines." 
The  frequency  of  these  machines  was  from  40  to  50. 

Alternators  with  pole  armatures  of  the  Ganz  type, 
giving  a  frequency  of  42,  are  frequently  worked  in 
parallel  in  European  plants.  In  the  plant  at  Rome  it 
has  been  found  possible  to  operate  Ganz  alternators  of 
different  sizes  together.* 

Steinmetz  has  operated  alternators  of  the  General 
Electric  Company  in  parallel,  under  the  following  con- 
ditions :  f 

Two  60  K.W.  alternators  with  toothed  armature 
cores,  giving  a  frequency  of  125,  were  experimented 
upon.  These  were  first  excited  so  as  to  give  a 
pressure  of  1000  volts.  The  machines  were  then 
switched  into  parallel  without  making  any  effort  to 
first  get  them  into  step.  They  quickly  dropped  into 
step,  and  ran  synchronously  with  an  interchange  of 
wattless  current  of  only  four  amperes.  Since  the  nor- 
mal full  load  of  these  machines  was  52  amperes  at 
1 1 50  volts,  this  is  a  remarkably  good  result.  Experi- 
ments were  then  made  to  determine  the  momentary 
rush  of  current  at  the  instant  when  the  machines  were 


*  Fleming,  Alternate  Current  Transformer,  Vol.  II.,  p.  134;   Hedges, 
Continental  Electric  Lighting  Stations,  p.  14. 

t  Parallel  Running  of  Alternators,  Electrical  World,  Vol.  23,  p.  285. 


REGULATION   AND   COMBINED   OUTPUT.        347 


thrown  together,  under  various  conditions  of  phase. 
To  determine  the  phase  relations,  a  synchronizer  was 
used.  The  machines  were  first  brought  to  equality  of 
pressure  (1000  volts)  and  synchronism,  and  then  approx- 
imately into  step.  They  were  then  switched  together. 
The  momentary  rush  of  current  was  from  .5  to  6  am- 
peres greater  than  the  regular  wattless  synchronizing 
current,  and  depended  in  magnitude  upon  the  care  taken 
to  bring  the  machines  into  exact  step  before  they  were 
thrown  together.  The  machines  were  then  thrown  to- 
gether, when  their  phases  were  180°  from  step,  so  that  the 
machines  would  act  in  series  on  short-circuit  instead  of  in 
parallel.  When  the  switch  was  closed,  a  large  instanta- 
neous current  passed  through  the  machines  for  a  fraction 
of  a  second,  and  the  machines  came  at  once  into  step. 

Mr.  Steinmetz  then  made  experiments  upon  the  action 
of  the  machines  when  thrown  in  parallel  with  their 
voltages  different.  The  results  are  given  in  the  follow- 
ing table  and  the  accompanying  figure  (Fig.  162)  : 


A.  Machine 
Pressure. 

B.  Machine 
Pressure. 

A-B. 

Resultant 
Pressure. 

Synchronizing 
Current. 

Phasing 
Current. 

IOOO 

IOOO 

O 

996 

4.0 

2.O 

IIOO 

900 

200 

IOOO 

6.5 

•5 

1200 

800 

400 

IOOO 

I3.0 

3-° 

1300 

700 

600 

IOOO 

18.0 

4.0 

I4OO 

600 

800 

1026 

24.0 

6.0 

I5OO 

500 

IOOO 

IOIO 

28.0 

6.0 

I6OO 

400 

I2OO 

1040 

39-o 

3-0 

lyOO 

300 

I4OO 

1046 

44.0 

3-o 

I800 

200 

I6OO 

1060 

50.0 

6.0 

IQOO 

100 

I800 

1066 

56.5 

5-5 

2OOO 

O 

2000 

1075  ±70 

62.0 

10.0 

348 


ALTERNATING  CURRENTS. 


The  meaning  of  the  first  four  columns  is  evident  from 
the  headings,  the  fifth  gives  the  synchronizing  current 
necessary  to  hold  the  machines  in  step,  and  the  sixth  the 
additional  current  that  flows  between  the  machines  for 
an  instant  when  they  are  first  thrown  together  and  are 
out  of  step.  When  the  difference  in  the  pressures  of 


18          24         30          36          42 

SYNCHRONIZING   CURRENT 

3456  7 

PHASING  CURRENT 


Fig.  162 

the  two  machines  became  1900  volts,  the  resultant 
pressure  began  to  be  unsteady.  When  the  difference 
was  2000  volts,  the  resultant  pressure  varied  70  volts 
on  either  side  of  1075.  The  irregularity  of  the  curve  of 
phasing  current  may  be  due  to  the  differences  in  the 
relative  phases  of  the  machines  at  the  instants  when 
they  were  thrown  together. 


REGULATION   AND    COMBINED   OUTPUT.        349 

In  discussing  these  experiments  Mr.  Steinmetz  says  : 
"We  may  discard  all  the  usual  theoretical  statements 
on  parallel  working  relating  to  the  effect  of  frequency, 
self-induction,  etc.,  as  wholly  disproved  by  experience. 
.  .  .  With  regard  to  frequency,  I  investigated  the 
parallel  working  of  alternators  at  a  frequency  as  high 
as  125  cycles,  and  at  a  frequency  as  low  as  25  cycles 
per  second,  and  found  no  difference  whatever ;  and  at 
the  high  frequency,  as  well  as  at  very  low  frequency, 
machines  properly  designed  for  these  frequencies  work 
perfectly  in  synchronism." 

From  all  the  evidence  thus  presented,  it  may  reason- 
ably be  concluded  that  frequency,  within  the  limits  of 
common  practice,  is  not  an  element  affecting  the  suc- 
cess of  parallel  working  of  alternators.  This  is  in 
full  accord  with  the  deductions  of  Mordey  and  Stein- 
metz.* 

92.  The  Effect  of  Armature  Inductance  on  Parallel 
Operation.  —  Successful  parallel  operation  of  alternators 
depends  upon  their  holding  each  other  in  synchronism 
and  step,  even  when  the  prime  movers  do  not  naturally 
synchronize.  The  effort  of  the  machines  to  do  this  is  the 
fundamental  cause  for  the  flow  of  a  wattless  synchroniz- 
ing current.  The  total  synchronizing  current  flowing 

*  Compare  Mordey,  Alternate  Current  Working,  Jour.  Inst.  E.  £., 
Vol.  18,  p.  591;  Snell,  The  Distribution  of  Power  by  Alternate-Current 
Motors,  Jour.  Inst.  E.  E.,  Vol.  22,  p.  280;  Mordey,  Testing  and  Work- 
ing Alternators,  Jour.  Inst.  E.  E.,  Vol.  22,  p.  116;  Mordey,  On  Parallel 
Working  with  special  reference  to  Long  Lines,  Jour.  Inst.  E.  E.,  Vol.  23; 
Forbes,  Electrical  Transmission  of  Power  from  Niagara  Falls,  Jour.  Inst. 
E.  E.,  Vol.  22,  p.  484;  and  the  discussions  on  these  papers;  also  Stein- 
metz, Parallel  Running  of  Alternators,  Electrical  World,  Vol.  23,  p.  285; 
C.  E.  L.  Brown,  Jour.  Inst.  E.  E.,  Vol.  22,  p.  600. 


350  ALTERNATING   CURRENTS. 

between  a  machine  and  the  station  'bus  bars  may  be 
resolved  into  two  components,  one  in  phase  with  the 
pressure  which  causes  the  synchronizing  current  to 
flow,  and  the  other  lagging  90°  behind  the  pressure. 
The  former  is  usually  quite  small.  Thus  suppose  in 
Fig.  163  that  b  is  the  curve  of  pressure  of  a  machine, 
and  a  the  curve  for  the  station  'bus  bars.  As  a  and 
b  are  slightly  out  of  opposition,  there  will  be  a  re- 
sultant pressure,  q,  tending  to  set  up  a  cross  or  series 
current.  This  current  may  be  considered  as  made  up 
of  a  component  s,  in  phase  with  q,  and  of  another 
component  ?/,  in  quadrature  with  q,  the  former  being 
the  active,  and  the  latter  the  wattless  component.  The 
component  s  has  the  same  effect  upon  both  a  and  b  dur- 
ing a  complete  period,  hence  it  can  have  no  effect  in 
tending  to  draw  the  machine  into  phase  with  the  'bus 
bar  pressure ;  but  the  wattless  component  //,  which  is 
dependent  upon  the  self-inductance  of  the  series  cir- 
cuit, must,  from  its  position,  cause  a  motor  action  on 
the  lagging  machine  (b),  and  a  corresponding  genera- 
tor action  on  the  leading  machines  connected  to  the 
'bus  bars ;  and  it  thus  tends  to  draw  the  machines  into 
synchronism  (Sect.  86),  for  it  will  be  noticed  by  refer- 
ence to  the  figure  that  u  is  in  phase  with  m  and  in 
opposition  to  n,  and  that  m  and  n  are  respectively  the 
components  of  a  and  b,  which  are  in  opposition,  and 
therefore  working  on  the  parallel  circuit.  If  the  ma- 
chine, b,  led  the  'bus  bar  curve,  then  the  synchronizing 
current  would  retard  instead  of  assisting  it,  as  the 
figure  plainly  shows.  The  effect  of  the  synchronizing 
current  in  dragging  the  machines  into  step  depends 


REGULATION   AND    COMBINED   OUTPUT.        351 


\ 


352  ALTERNATING    CURRENTS. 

upon  its  magnitude  and  relative  phase,  while  its  magni- 
tude depends  :  (i)  upon  the  algebraic  sum,  or,  what  is 
the  same  thing,  the  arithmetical  difference  between  the 
instantaneous  pressure  at  the  'bus  bars  and  the  instan- 
taneous pressure  developed  by  the  machine  ;  (2)  upon 
the  reciprocal  of  the  impedance  of  the  machine  arma- 
ture. In  other  words,  the  instantaneous  synchronizing 
current  flowing  through  any  one  machine  which  is 
connected  to  'bus  bars  is 


_ 

= 


where  ea  and  e^  are  the  instantaneous  pressures  of  the 
'bus  bars  and  the  armature,  which  are  always  of  oppo- 
site sign  and  equal  when  the  machines  are  in  exact  syn- 
chronism and  step,  and  Ra  and  La  are  respectively  the 
resistance  and  the  inductance  of  the  armature  circuit 
including  the  leads  from  the  'bus  bars.  It  is  here 
assumed  that  the  effective  pressure  at  the  'bus  bars 
and  that  developed  by  the  machine  are  equal,  which  is 
an  essential  condition  for  the  synchronizing  current  to 
be  practically  wattless,  and  is  the  condition  in  which 
the  machines  are  run  in  practice.  Now  suppose  the 
pressure  curve  of  the  machine  under,  consideration  to 
lag  behind  the  phase  of  the  pressure  curve  at  the 
'bus  bars  by  an  angle  fi.  Let  the  effective  values  of 
these  pressures  be  E,  and  the  corresponding  maximum 
pressure  be  em.  At  any  moment  the  instantaneous 
pressures  are  ea  and  eb)  and 

ea  =  em  sin  a  =  A/2  E  sin  a, 
eb  =  em  sin  (a  +  180°  -  ft)  =  V2  E  sin  (a  +  180°  -  ft). 


REGULATION   AND    COMBINED   OUTPUT.        353 

The  instantaneous  pressure  causing  a  synchronizing 
current  to  flow  is  the  algebraic  sum  of  these,  or 

ea  +  eb  =  V2  E  [sin  a  +  sin  (a  +  180°  -  0)]. 

It  is  evident  from  the  figure  (Fig.  163)  that  this  is  a 
maximum  when  ea  and  et  are  equal  and  of  similar  signs, 
in  which  case  a  =  £  /3.  Then  the  maximum  value  of 
the  pressure  causing  a  synchronizing  current  is  found 
by  substituting  this  value  of  a  in  the  expression  for 
ea  +  eb,  and  (ea  +  cb)m  =  2  A/2  E  sin  \$. 

The  effective  value  of  the  pressure  is  then  2  E  sin  J  0, 
and  the  synchronizing  current  is 

2  E  sin     3     * 


„_ 

= 


For  smooth  and  successful  working  in  parallel,  C,  must 
become  sufficiently  great,  in  case  the  machines  tend  to 
get  out  of  step,  to  pull  the  machines  together  before  & 
becomes  of  appreciable  magnitude.  Hence  it  is  neces- 
sary that  the  denominator  in  the  expression  for  Ct  be  as 
small  as  possible.  In  other  words,  armature  impedance 
must  be  as  small  as  possible.  Figure  163  shows  plainly 
that  the  pressure  (Oq)  causing  Ct  is  behind  the  phase  of 
the  machine  pressure  by  an  angle  90°  —  -|  /3.  The  syn- 
chronizing current  (Ct  =  Oc]  lags  behind  the  pressure  by 
an  angle  c/>4  (qOc),  the  tangent  of  which  is  —  ^  —  -. 

•Ra 

Consequently  the  synchronizing  current  is  out  of  phase 
with  the  machine  pressure  by  an  angle  of  90°  +  (<£,—  -J-yS). 
A  current  (wattless)  which  has  a  phase  difference  of 
1  80°  or  o°  compared  with  the  pressure  of  a  machine  has 
the  strongest  effect  in  bringing  the  machine  into  step. 
2  A 


354  ALTERNATING   CURRENTS. 

This  effect  is  to  retard  or  accelerate  the  refractory  ma- 
chine depending  on  whether  the  phase  difference  is 
o°  or  1  80°,  which  in  turn  depends  upon  whether  the 
machine  is  leading  or  trailing.  The  action  in  case  of  a 
leading  machine  would  be  represented  by  the  left-hand 
half  of  Fig.  163  if  it  were  reversed. 

The  wattless  component    (C^  of   the  synchronizing 
current  just  found  is, 

C^  =  Cs  sin  <£s, 
and  therefore, 

_        2  E  sin  \  /3  2?r/Za  _ 

~  '  ' 


=2sn-^ 

and,  with  a  fixed  value  of  Ra,  this  will  have  a  maximum 
value  when 

Rc  =  27rfLa  or  when  tan  <£s  =  i  and  ^  =  45°. 
The  maximum  possible  value  of  C^  is  therefore 

~  2Esmlj3 

c  max  =  -  ' 


and  the  corresponding  value  of  Cs,  the  total  synchroniz- 
ing  current,  is 


The  limits  in  the  value  of  Ra  are  fixed  by  considera- 
tions of  economy  in  construction  and  of  efficiency,  and 
the  frequency  is  fixed  by  conditions  of  operation  ;  La 
is  therefore  the  only  independent  variable  in  the  pre- 
ceding equations.  In  order  to  have  the  most  sensitive 
mutual  control,  the  self-inductance  of  the  armature 
circuit,  which  at  its  least  value  is  always  many  times 


REGULATION   AND   COMBINED   OUTPUT.        355 


r> 

larger  than  — -,  must  be  as  small  as  possible.      If  it 

27T/ 

were  possible  to  reduce  the  reactance  to  the  value  of 
the  resistance,  the  jerking  of  a  refractory  alternator  into 
phase  would  probably  be  too  severe  for  good  working, 
but  in  commercial  machines  such  trouble  is  not  likely  to 
exist  on  account  of  the  unavoidable  magnitude  of  the 
self-inductance,  which  cannot  be  reduced  beyond  certain 
limits.  The  correctness  of  the  formulas  thus  deduced 
has  not  been  studied  experimentally,  but  it  might  be 
readily  investigated  with  the  aid  of  Bedell's  ingenious 
phase  indicator.*  f 

*  Bedell,  Trans.  Amer.  Inst.  E.  E.,  Vol.  n,  p.  502. 

f  Diagrams  I  and  2  given  herewith,  which  are  similar  in  plan  to  those 
given  in  the  footnote  on  page  326,  show  the  conditions  of  synchronizing 
very  well.  Figure  I  applies  to  an  alternator  which  lags  behind  its  proper 
phase,  and  Fig.  2  to  one  which  leads. 


FIG.  2 


OA  represents  the  'bus  bar  pressure  in  phase  and  magnitude. 
OB  represents  the  machine  pressure  in  phase  and  magnitude. 
OR  represents  the  resultant,  or  synchronizing,  pressure  in  phase  and 
magnitude. 


356         ALTERNATING  CURRENTS. 

The  list  of  examples  of  parallel  working  (Sect.  91)  con- 
tains machines  having  armatures  of  very  different  resist- 
ances and  inductances.  The  Westing-house,  Thomson- 
Houston,  and  Elwell-Parker  machines  have  smooth  iron 
armature  cores  and  fairly  low  armature  inductances  and 
resistances.  The  armature  inductance  of  the  Mordey 
and  Ferranti  machines  is  probably  somewhat  smaller, 
though  entirely  comparable  with  these  values.  The 
Kapp  machines,  and  the  General  Electric  machines 
with  which  Steinmetz  experimented,  have  much  greater 
armature  inductances,  and  the  armatures  of  the  Stanley 
inductor  machines  probably  have  inductances  of  inter- 
mediate values.  All  these  machines  have  been  shown 
to  run  in  parallel  with  similar  machines  with  fair  satis- 
faction, while  Mordey,  Steinmetz,  and  Stanley  have 

Oc  represents  the  synchronizing   current  in  phase  and  magnitude. 

Ou  represents  its  wattless,  or  phasing,  component. 

/3  is  the  angle  by  which  the  machine  differs  from  its  proper  phase  for 
parallel  working,  OB1 . 

0S  is  the  angle  c Os  by  which  the  synchronizing  current  lags  behind 
the  resultant  pressure. 

The  figures  show  plainly  that  as  /3  increases  OR  increases,  and  at  the 
same  time  Oc  and  Ou  increase.  The  product  OB  X  OK  X  cos  £  /3  is 
proportional  to  the  synchronizing  torque  exerted  on  the  machine;  it  is 
negative  in  Fig.  I  and  positive  in  Fig.  2,  so  that  the  torque  is  exerted  on 
the  machine  and  accelerates  it  in  one  case,  and  it  is  exerted  by  the  machine 
and  retards  it  in  the  other  case.  For  any  given  value  of  j3,  the  torque  is  a 
maximum  when  Ou  X  OB  is  a  maximum,  which  occurs  when  2  irfL  =  A, 
or  <ps  =  45°.  In  a  1000  volt  50  K.W.  machine  giving  a  frequency  of  60 
and  having  a  resistance  in  the  armature  circuit  of  .5  ohm,  it  is  required 
that  L  =  .0013  to  give  a  maximum  synchronizing  effect,  which  is  many 
times  smaller  than  the  smallest  value  which  is  commercially  attainable  in 
any  type  of  alternator. 

An  inspection  of  the  figures  shows  that  if  the  synchronizing  current  led 
the  resultant  pressure  there  would  be  no  tendency  for  the  machine  to  come 
into  parallel  with  the  'bus  bars. 


REGULATION   AND  COMBINED   OUTPUT.        357 

experimentally  shown  that  Mordey  disc  armature  ma- 
chines, the  General  Electric  ironclad  armature  machines, 
and  Stanley  toothed  armature  inductor  two-phase  ma- 
chines will  run  parallel  with  machines  of  their  own  type 
with  excellent  results.  On  the  other  hand,  Gordon  has 
said  that  his  machines  did  not  come  into  step  well 
(page  345),  and  Hopkinson's  experiments  show  that 
De  Meritens  machines  take  an  excessive  synchroniz- 
ing current.  The  armatures  of  both  the  Gordon  and 
De  Meritens  alternators  have  an  excessive  resistance 
and  self-inductance  as  compared  with  good  machines 
of  the  present  day,*  and  their  actions  fully  agree  with 
the  indications  of  the  formulas  already  given. 

We  are  therefore  entirely  justified  in  drawing  the 
conclusion  that  in  machines  intended  for  parallel  work- 
ing both  armature  resistance  and  inductance  should  be 
small,  but  that  within  the  range  of  resistances  and 
inductances  to  be  found  in  modern  commercial  machines 
of  economical  design  the  armature  inductance  is  too 
small  to  have  a  detrimental  effect  upon  parallel  work- 
ing though  it  is  much  too  large  to  give  a  maximum  syn- 
chronizing torque,  which  in  fact  may  be  an  advantage, 
as  it  saves  the  machines  from  being  torn  to  pieces. 
Some  types  of  modern  alternators  of  less  economical 
design  may  have  too  much  armature  self-inductance  to 
operate  perfectly  in  parallel. 

These  deductions  are  strengthened  by  the  experimen- 
tal results  gained  by  Mordey  and  Steinmetz.  In  dis- 
cussing his  experiments  already  summarized  (page  343) 

*  Gordon's  Electric  Lighting,  pp.  156,  162;  London  Electrician^  Vol. 
'7.  P- 5'- 


358         ALTERNATING  CURRENTS. 

Mordey  says  :  "  It  may  be  pointed  out  that  these  tests 
were  made  under  the  most  exacting  and  onerous  condi- 
tions that  could  possibly  be  imposed,  and  particularly  I 
would  point  out  that  on  account  of  the  very  great 
momentum  of  the  revolving  masses  nothing  but  the 
strongest  and  most  instantaneous  motor  action  could 
have  kept  the  machines  in  place.  There  never  was  a 
single  case  when  they  got  out  of  step,  even  momentarily 
or  when  subjected  to  sudden  and  violent  variations  of 
load.  When  it  is  considered  that  in  order  to  secure 
this  result,  it  was  imperative  that  the  control  of  all  that 
mass  should  be  exerted  in  a  fraction  of  ^^  of  a  second 
(the  periodicity  being  100),  it  will  be  recognized  that 
there  was  no  time  to  be  lost,  and  that  the  use  of  any 
self-induction  or  resistance,  or  of  anything  else  that 
could  in  any  way  choke,  retard,  check,  or  interfere  with 
the  strength  and  instantaneity  of  the  action,  was  above 
all  things  to  be  avoided. 

"  I  should  mention  that  the  machines  apparently 
synchronized  equally  well  at  speeds  varying  very  con- 
siderably. 

"  As  to  the  self-induction  of  the  machine  itself,  that 
is  quite  negligible.  Its  characteristic  (Fig.  164)  is 
nearly  straight,  about  half  the  drop  in  the  curve  being 
due  to  resistance  and  half  to  self-induction."  * 

Steinmetz  says  in  regard  to  his  experiments  :  "  The 
self-induction  of  machines  within  the  limits  met  in  well- 
designed  alternators  has  nothing  to  do  with  the  success 
of  synchronous  running,  and  I  had  three  phasers  of 
very  low  armature  self-induction  running  in  parallel 

*  London  Electrician,  Vol.  23,  p.  66. 


REGULATION   AND   COMBINED    OUTPUT.        359 

with  each  other  and  ironclad  (imbedded  armatures)  sin- 
gle-phasers  of  high  armature  self-induction."  * 

O  1^  O 

The   comparatively   high   self-inductance    of   ironclad 
alternator  armatures  has   proved    advantageous  in   an- 


2,500 


2,000 


1,500 

co 
h- 

O 

1,000 


500 


10 

AMPERES 
Fig-.  164 


15 


other  direction,  that  of  saving  the  machine  from  the 
damaging  effects  of  sharp  short-circuits.  A  sharp 
short-circuit  upon  a  machine  with  a  disc  armature  is 
likely  to  cause  an  enormous  momentary  rush  of  current 


*  Electrical  World,  Vol.  23,  p.  285. 


ALTERNATING   CURRENTS. 

which  may  damage  or  destroy  the  armature,  while  the 
higher  self-inductance  of  the  ironclad  armature  chokes 
down  the  current  to  a  considerable  extent.  This  qual- 
ity of  the  ironclad  alternator  is  an  advantage  in  power 
transmission  plants  or  small  electric  light  plants,  where 
little  stress  is  laid  upon  exact  regulation,  and  the  sta- 
tion and  lines  are  poorly  cared  for.  In  large  and  well- 
operated  electric  light  stations,  the  injurious  effect  of 
self-inductance  upon  the  regulation  of  the  pressure  is 
of  preponderating  importance ;  and  since  it  is  of  the 


Fig.  165 

utmost  moment  that  alternators  used  in  large  electric 
light  stations  shall  have  -the  least  possible  defect  in 
regulation,  the  alternator  with  large  self-inductance  is 
not  so  well  adapted  to  this  service  (Sects.  67  and  74). 

93.    Effect  of   the  Form   of  the  Pressure   Curve  and 
of  Armature  Reactions   on  Parallel  Operation.  —  Little 


REGULATION   AND  COMBINED   OUTPUT.        361 

experimental  evidence  is  available  bearing  upon  this 
question.  The  able  designer,  C.  E.  L.  Brown,  states 
that  he  has  operated  his  own  alternators  with  smooth 
iron  armature  cores  in  parallel  with  Ganz  alternators 
which  have  the  usual  pole-type  armatures.*  The  former 
machines  give  a  pressure  curve  which  approximates  a 
sinusoid,  while  the  curve  given  by  the  latter  is  quite 
irregular  and  peaked  (Fig.  165).  Mr.  Steinmetz  has 
operated  machines  with  smooth  iron  armature  cores  in 


Fig.  166 

parallel  with  others  having  toothed  cores,  f  It  is  not  re- 
lated how  far  the  pressure  curves  of  these  machines  dif- 
fered, but  probably  some  differences  existed.  These 
statements  seem  to  show  that  machines  with  different 
pressure  curves  can  be  run  together  satisfactorily,  but 
they  undoubtedly  require  a  considerably  increased  ex- 
change of  current  as  compared  with  the  synchronizing 
current  of  machines  which  give  curves  which  are  exactly 

*  Jour.  Inst.  E.  E.,  Vol.  22,  p.  600. 
t  Electrical  World,  Vol.  23,  p.  285. 


362         ALTERNATING  CURRENTS. 

alike.  For,  since  the  unlike  curves  cannot  coincide  even 
when  the  machines  are  in  exact  synchronism,  a  current 
of  more  or  less  irregular  form  must  be  exchanged  by 
the  machines,  and  is  therefore  superposed  upon  the  true 
synchronizing  current.  This  superposed  current  may 
have  a  very  different  frequency  from  that  of  the  ma- 
chines (Fig.  1 66)  and  is  not  necessarily  wattless. 

94.  The  Effect  on  Parallel  Operation  of  Regulation 
by  Varying  the  Excitation.  —  Suppose  that  a  Grove's 
and  a  Daniell's  cell  be  so  constructed  that  their  inter- 


GROVE  CELL 
1.9  Volts 


DANIELL     CELL 
1.1  Volts 

Fig.  167 


nal  resistances  are  equal.  Their  pressures  may  be 
assumed  to  be  approximately  1.9  and  i.i  volts.  If 
these  cells  are  connected  in  parallel  (Fig.  167)  to  a 
circuit,  the  pressure  impressed  upon  the  circuit  is  the 
mean  of  the  pressure  developed  by  the  cells,  or  1.5 
volts.  This  is  brought  about  as  follows  :  The  Daniell 
cell  serves  as  a  shunt  across  the  terminals  of  the  Grove 
cell,  so  that  current  from  the  Grove  cell  has  two  paths, 
one  through  the  external  circuit  and  one  backwards 
through  the  Daniell  cell.  Sufficient  current  will  flow 


REGULATION   AND   COMBINED   OUTPUT.        363 

through  the  latter  to  equalize  the  pressure  at  the  ter- 
minals by  means  of  the  fall  .of  pressure  over  the  internal 
resistances  of  the  cells.  If  the  external  circuit  is  open, 
a  current  will  flow  through  the  cells  which  is  sufficient 
to  cause  a  loss  of  pressure  in  each  cell  of  .4  of  a  volt. 
The  pressure  at  the  terminals  of  the  Grove  cell  is  then 
1.9  — .4=  1.5,  and  at  the  terminals  of  the  Daniell  cell 
i.i  +.4=  1.5.  Suppose  the  external  circuit  is  closed 
and  demands  enough  current  to  cause  a  pressure  loss  in 
the  Grove  cell  of  .2  of  a  volt.  Then  sufficient  addi- 
tional current  will  flow  through  the  circuit  of  the  cells 
to  make  the  terminal  pressure  1.7  — .3  =  1.4  and  i.i 
-f  .3  =  1.4.  From  an  extension  of  this  reasoning  it  is 
seen  that  a  current  flows  backwards  through  the  Daniell 
cell  under  all  conditions  of  the  circuit  until  sufficient 
current  is  demanded  by  the  external  circuit  to  make 
the  pressure  loss  in  the  Grove  cell  .8  of  a  volt.  Then 
the  Daniell  cell  is  exactly  balanced,  and  will  furnish 
half  of  any  additional  current  demanded  by  the  external 
circuit. 

The  same  condition  of  affairs  exists  when  continuous- 
current  dynamos  operate  in  parallel.  If  their  pressures 
are  not  exactly  equal,  the  'bus  bar  pressure  will  be  a 
mean  value.  The  machine  of  lower  pressure  will  sup- 
ply sufficiently  less  current  so  that  an  equalization  of 
pressure  comes  about  through  changes  in  the  loss  of 
pressure  due  to  the  resistances  of  the  armature  circuits 
and  the  effects  of  armature  reactions.  If  the  low- 
pressure  machine  falls  behind  too  much,  the  current 
through  its  armature  will  be  reversed  (Vol.  I.,  p.  231). 
The  inequality  of  load  can  be  readily  perceived  in  the 


364         ALTERNATING  CURRENTS. 

case  of  continuous-current   dynamos  by  observing  the 
amperemeters. 

In  the  case  of  alternators,  similar  conditions  exist. 
For  illustration  we  will  assume  the  machines  to  give 
sine  pressure  curves  which  are  in  synchronism  and 
step.  If  the  machines  give  equal  pressures,  they  will 
feed  the  external  circuit  equally  if  they  are  of  the  same 
capacity,  or  proportionally  if  they  are  of  unequal  capac- 
ities. If  one  machine,  which  is  connected  to  constant- 
pressure  'bus  bars,  gives  a  pressure  which  is  too  low, 
it  will  not  furnish  its  share  of  current  to  the  circuit. 
The  pressure  at  the  'bus  bars  is  equalized  through  the 
influence  of  the  pressure  losses  caused  by  the  impe- 
dances of  the  various  armature  circuits.  If  the  press- 
ure of  the  machine  becomes  much  lower  than  that  of 
the  'bus  bars,  a  reverse  current  will  flow  through  the 
machine,  and  it  will  run  as  a  motor.  This  is  not  shown 
by  the  amperemeter,  as  in  the  case  of  continuous-cur- 
rent machines.  It  may  be  observed,  however,  by  alter- 
ing the  excitation.  If  the  machine  is  doing  its  work 
properly,  decreasing  the  excitation  will  decrease  its 
apparent  output  to  a  minimum,  after  which  further 
decrease  in  the  excitation  causes  an  increase  in  the 
apparent  output.  The  machine  is  then  receiving  more 
energy  from 'the  'bus  bars  than  it  returns.  In  order, 
then,  that  machines  working  in  parallel  may  be  relied 
upon  to  contribute  energy  to  the  external  circuit,  it  is 
necessary  that  they  be  so  sufficiently  excited  that  an 
increase  in  the  excitation  will  cause  an  increase  in  the 
apparent  output.  This,  of  course,  is  exactly  what  is 
done  with  continuous-current  dynamos,  using  the  indi- 


REGULATION   AND   COMBINED    OUTPUT.        365 

cations  of  the  amperemeter  as  a  guide,  but  in  the  case 
of  alternators,  the  amperemeters,  alone,  fail  to  show 
what  the  machines  are  really  doing.  This  is  partly  on 
account  of  their  inability  to  indicate  the  direction  of 
the  transfer  of  energy  in  an  alternating  circuit,  and 
partly  on  account  of  the  masking  effect  of  a  wattless 
current  ;  but  if  direct-reading  wattmeters  are  used  in 
place  of  amperemeters  in  the  circuits  of  the  alternators, 
the  indications  of  these  instruments  may  be  used  as  a 
guide  in  handling  the  machines  when  in  parallel,  exactly 
as  the  indications  of  amperemeters  are  used  as  a  guide 
for  handling  continuous-current  dynamos  which  are  con- 
nected in  parallel. 

When  two  alternators  of  unequal  pressure  are  con- 
nected together,  there  is  quite  a  curious  result.  The 
terminal  pressure  is  a  mean  between  the  machine 
pressures  and  the  machines  run  together  without  a 
dangerous  exchange  of  current*  The  instantaneous 
value  of  the  current  exchanged  by  the  two  machines 
at  any  moment,  on  the  supposition  that  the  machines 
are  in  step  and  that  the  pressure  curves  are  sinusoids, 
is 

-\/2  E1  sin  a  —  A/2  E"  sin  a  _  ^/~2(E'  —  E"}  sin  a  . 

~ 


and  its  effective  value  is 

E'  -  E" 


where  E'  and  En  are  the  effective  pressures  developed 

*  Mordey's  and  Steinmetz's  Experiments,  Section  91. 


366         ALTERNATING  CURRENTS. 

by  the  two  machines,  and  R  and  L  are  the  resistance 
and  inductance  of  the  two  armature  circuits  taken 
in  series.  Take  two  50  K.W.  alternators  designed  to 
give  50  amperes  at  1000  volts  with  a  frequency  of  125, 
assuming  the  armature  resistance  and  inductance  of 
each  to  be  respectively  .45  ohm  and  .015  henry.  Then 
if  the  two  machines  be  thrown  in  parallel  with  one 
excited  to  give  1000  volts  and  the  other  without 
excitation,  the  current  exchanged  amounts  to  about 
42  amperes,  which  is  less  than  the  full  load  of  the 
machines ;  while  if  an  unexcited  machine  be  connected 
to  'bus  bars  of  a  large  system,  the  current  sent  through 
its  armature  only  amounts  to  85  amperes.  The  current 
that  would  be  exchanged  under  similar  conditions  by 
continuous-current  machines  would  be  over  uoo  am- 
peres, which  would  immediately  ruin  the  machines. 
The  difference  in  the  two  results  is  due  to  the  induc- 
tive effects  of  the  alternating  current.  If  the  two 
alternators  have  a  considerable  armature  reaction,  the 
difference  is  likely  to  be  still  more  marked,  since  arma- 
ture reactions  tend  to  weaken  an  alternator's  field  when 
running  as  a  generator,  and  strengthen  it  when  running 
as  a  motor  '(Sect.  70). 

95.  The  Effect  of  Irregular  Angular  Velocity  on  Par- 
allel Operation. — The  angular  velocity  of  the  steam 
engines,  which  are  ordinarily  used  for  driving  dynamos, 
is  by  no  means  uniform  throughout  the  stroke,  though 
the  strokes  may  be  entirely  isochronous.  Curves  show- 
ing the  instantaneous  crank  velocities  taken  from  en- 
gines of  well-known  types  are  often  quite  irregular.  If 
alternators  are  driven  from  separate  engines,  this  irreg- 


REGULATION   AND   COMBINED   OUTPUT.        367 

ularity  of  angular  velocity  may  be  the  cause  of  more  or 
less  difficulty  in  parallel  working,  and  some  machines 
may  work  quite  satisfactorily  in  parallel  when  driven 
from  the  same  countershaft  or  from  separate  turbines 
(which  give  a  uniform  angular  velocity),  when  they  will 
not  do  so  if  driven  by  separate  engines. 

96.  Final  Conclusions  on  Parallel  Operation. —  i.  There 
is  no  difficulty  in  operating  alternators  of  good  commer- 
cial design  in  parallel. 

2.  Within  the  limits  of  practice  frequency  does  not 
materially  affect  parallel  working. 

3.  This  being  accepted,  smooth  parallel  working  re- 
quires that  the  armature  impedance  be  reduced  to  a 
minimum    (Sect.   92).     Armature   resistance   must   de- 
pend upon  considerations  of  economy. 

4.  Machines  with  excessively  large  armature  induct- 
ance will  run  in  parallel,  but  are  likely  to  "  hunt." 

5.  Machines  with  pressure  curves  of  quite  different 
forms  will  run  together  in  parallel,  but  with  a  contin- 
uous interchange  of  current,. the  magnitude  of  which 
depends  upon   the  value   of  the  armature  impedances 
and  the  relative  form  of  the  pressure  curves. 

6.  The  dynamo  amperemeters  which  are  ordinarily 
used  should  be  replaced,  or  reinforced,  when  machines 
are  run  in  parallel,  by  direct  reading  wattmeters.     The 
excitation  of  the  machines  may  then  be  adjusted  so  that 
the  wattmeters  show  a  proper  division  of  the  load.     In 
that  case  amperemeters  in  the  dynamo  circuits  should 
show  an  approximately  similar  division  of  the  current. 

7.  If  the  excitation  of  the  machines  is  properly  ad- 
justed the  sum  of  the  readings  of  the  machine  watt- 


368  ALTERNATING   CURRENTS. 

meters  will  equal  the  reading  of  a  main  wattmeter  in 
the  main  circuit  of  the  'bus  bars,  and  the  sum  of  the 
readings  of  the  machine  amperemeters  will  be  a  little 
greater  than  the  reading  of  the  main  amperemeter  (the 
difference  being  twice  the  value  of  synchronizing  cur- 
rent exchanged  by  the  machines). 

8.  The   effect   of   inductance   makes    such    an   over- 
whelming part  of  the  impedance  of  even  the  best  com- 
mercial machines,  that  an  injuriously  excessive  current 
is    not   likely  to  flow  through   a   machine    even   when 
switched  onto  the  'bus  bars  entirely  unexcited.      (This 
is  true  to  a  marked  degree  of  machines  with  ironclad 
armatures.) 

9.  Parallel  working  is  likely  to  be  more  successful 
when  the  alternators  are  driven  from  the  same  engine 
or  counter  shaft,  or  when  the  prime  movers  are  tur- 
bines. 

These  conclusions  are  based  upon  the  operation  of 
alternators  which  are  driven  by  self -regulating  prime 
movers ;  that  is,  the  driving  power  transmitted  to  the 
alternators  is  automatically  adjusted  to  the  demands  of 
the  machines,  and  they  are  driven  at  a  uniform  speed 
regardless  of  their  load.  This  is  in  accord  with  the 
usual  American  practice  in  operating  dynamos.  In 
England  it  is  apparently  common  to  work  alternators 
from  engines  that  are  not  self-regulating  and  which 
furnish  a  fixed  amount  of  power  when  running  syn- 
chronously unless  the  position  of  the  throttle  valve  is 
altered.* 

*  Kapp's  Dynamos,  Alternators,  and  7^ransformers,  p.  399;  SnelPs 
Electric  Motive  Power,  p.  241. 


REGULATION   AND   COMBINED   OUTPUT.        369 

If  composite-wound  alternators  are  to  be  operated  in 
parallel,  it  is  necessary  to  use  an  equalizing  connection, 
exactly  as  in  compound-wound  continuous-current  ma- 
chines, to  equalize  the  effects  of  the  compounding 
(Vol.  L,  p.  237). 

2B 


370  ALTERNATING   CURRENTS. 


CHAPTER   VIII. 

EFFICIENCIES,    ETC. 
jp 

97.  General  Considerations. — The  general  definitions 
of  efficiencies,  which  have  already  been  explained  with 
regard  to  continuous-current  machines,  are  applicable 
to  alternators  (Vol.  I.,  p.  244).  The  principal  causes 
of  loss  in  alternators  are  the  same  as  those  in  contin- 
uous-current machines ;  but  on  account  of  increased 
frequencies,  the  effects  of  foucault  currents  and  hys- 
teresis are  intensified.  On  this  account  particular  care 
is  required  in  selecting  and  annealing  the  iron  for  the 
armature  core,  and  in  insulating  the  armature  discs 
from  each  other.  Advantage  should  also  be  taken  of 
every  opportunity  for  ventilation.  The  magnetic  density 
in  alternator  armature  cores  is  made  considerably  less 
than  in  continuous-current  armatures,  as  already  ex- 
plained (Sect.  61).  The  following  table  of  satisfactory 
densities  is  given  by  Snell.*  The  inductions,  which 
are  given  in  lines  of  force  per  square  centimeter  and 
per  square  inch,  may  be  assumed  as  fair  guides  for 
proper  judgment  in  designing. 

*  Electric  Motive  Power,  p.  175. 


EFFICIENCIES,    ETC. 


371 


TABLE  OF  ARMATURE   INDUCTIONS   SUITABLE  FOR 
VARIOUS   FREQUENCIES. 


Frequencies. 

B  (per  sq.  cm.)- 

B  (per  sq.  in.)- 

50 

5000-6200 

32,000-40,000 

60 

4650-5000 

30,000-32,000 

70 

4350-4650 

28,000-30,000 

80 

4000-4350 

26,000-28,000 

90 

3700-4000 

24,OOO-26,OOO 

100 

3500-3700 

22,500-24,000 

110 

3250-3500 

2I,OOO-22,5OO 

1  20 

3000-3250 

19,500-21,000 

130 

2800-3000 

18,000-19,500 

As  already  stated,  the  values  of  armature  inductions 
given  in  this  table  serve  excellently  as  a  guide  in  de- 
signing, but  they  are  frequently  exceeded  in  practice. 
It  is  no  uncommon  thing  to  find  inductions  as  great  as 
5000  to  7000  in  alternator  armatures,  giving  a  frequency 
of  125  or  more.  Snell's  table  is  apparently  modelled 
after  a  table  presented  by  Kolben  in  an  article  upon 
polyphase  induction  motors,  and  which  gives  the  range 
of  inductions  to  be  used  in  such  machines.  Kolben's 
table  is  given  in  Section  177. 

98.  Methods  of  Testing  Alternators.  —  The  experi- 
mental determination  of  the  efficiency  of  alternators 
may  be  made  by  methods  quite  analogous  to  those  used 
with  continuous-current  machines  (Vol.  L,  p.  255).  How- 
ever, instead  of  voltmeters  and  amperemeters  for  meas- 
uring electrical  energy,  wattmeters  must  be  used  (Fig. 
168),  unless  the  load  is  entirely  non-reactive.  In  case 


3/2  ALTERNATING   CURRENTS. 

a  transmission  dynamometer  is  used  to  measure  the 
power  absorbed  by  an  alternator,  the  commercial  effi- 
ciency is  equal  to  the  electrical  output,  determined  from 
the  readings  of  a  wattmeter,  divided  by  the  power 
shown  by  the  dynamometer  readings,  if  the  machine  is 
self-excited.  If  the  machine  is  separately  excited,  the 
energy  supplied  to  the  fields,  measured  by  amperemeter 
and  voltmeter  or  by  wattmeter,  must  be  added  to  the 
power  readings.  The  machine  friction  may  be  deter- 
mined from  the  dynamometer  readings  when  the  ma- 
chine is  run  with  its  fields  unexcited  and  the  external 
circuit  open.  The  total  loss  due  to  hysteresis  and 
foucault  currents  in  the  armature  core  and  conductors, 
may  be  approximately  determined  from  the  dynamome- 
ter readings  when  the  machine  is  operated  with  its  fields 
separately  excited  and  the  external  circuit  open,  and  the 
total  loss  may  be  approximately  separated  into  its  com- 
ponent parts  by  proceeding  in  an  entirely  analogous  way 
to  that  given  for  continuous-current  machines  (Vol.  I., 
p.  253,  First  Method}.  The  C2R  losses  in  fields  and 
armature  may  be  readily  computed.  If  the  machine  is 
self-excited  by  a  rectified  current,  the  field  current  will 
be  less  than  that  calculated  from  the  pressure  and 
resistance.  In  that  case  it  is  advisable  to  use  an  alter- 
nating-current amperemeter,  such  as  an  electrodyna- 
mometer,  to  determine  the  effective  current.  From  this, 
the  C^R  loss  may  be  computed  if  the  field  resistance  is 
known.  Or,  the  field  loss  may  be  directly  determined 
by  a  wattmeter  in  the  field  circuit. 

The  following  methods  are  especially  adapted  to  shop 
testing  and  determining  the  efficiency  of  an  alternator. 


EFFICIENCIES,    ETC. 


373 


EXCITER 


UUUUUUUJL 


WATT- 
METER 


I.  Modification  of  H op  kins  on  s  Method.  It  is  desir- 
able to  avoid  the  use  of  a  power  dynamometer,  and  with 
this  in  view  a  modification  of  Hop- 
kinson's  method  of  testing  (Vol. 
I.,  p.  256)  may  be  made  (Fig.  169). 
Two  equal  alternators  are  rigidly 
coupled  together  in  proper  step 
for  parallel  working.  Their  arma- 
tures are  electrically  connected  to- 
gether with  a  wattmeter  and  an 
electrodynamometer  in  the  circuit. 
The  fields  being  properly  excited 
by  a  separate  exciter,  so  that  one 
machine  will  act  as  a  generator 
and  the  other  as  motor,  the  system 
may  be  driven  by  supplying  suffi- 
cient power  to  make  up  the  ma- 
chine losses.  Assuming  the  arm- 
ature and  stray  losses  of  the  two 
machines  to  be  equal,  and  repre- 
senting the  wattmeter  reading  by  W,  the  power  sup- 
plied by  Py  and  the  resistance  of  the  connections  by  R^y 
then  the  efficiency  of  the  generator  is 

W 


Fig.  168 


where  c*Rf  is  the  field  loss  of  the  generator.  If  the 
machines  are  self-exciting,  the  power  in  the  field  circuits 
must  be  measured  and  proper  allowance  made.  The 
power  supplied  to  make  up  the  losses  may  be  measured 


374 


ALTERNATING   CURRENTS. 


by  a  transmission  dynamometer,  or  a  "rated"  contin- 
uous-current motor  may  be  used  to  supply  the  power. 
In  the  latter  case,  if  the  efficiency  of  the  motor  is 
known,  the  power  may  be  determined  by  measuring  the 
watts  supplied  to  the  motor  by  amperemeter  and  volt- 
meter. Or,  the  losses  of  the  motor  may  be  determined 


RHEOSTAT 


Fig.  169 

by  the  stray  power  method,  and  being  properly  deducted 
from  the  readings  of  power  absorbed  by  it  when  driv- 
ing the  alternators,  the  value  of  P  is  obtained  with 
sufficient  approximation.  The  three-machine  method 
might  be  directly  applied,  but  difficulties  due  to  syn- 
chronizing are  likely  to  appear. 


EFFICIENCIES,    ETC.  375 

2.  By  Rated  Motor.     Where  approximate  determina- 
tions  of   the  various   losses   of   conversion  and  of   the 
commercial  efficiency  are  sufficient,  the  alternator  tested 
may  be  driven  directly  from  a  "  rated  "  continuous-cur- 
rent motor.     By  means  of  amperemeter  and  voltmeter 
the  power  supplied  to   the  motor  may  be  determined 
with  the  alternator  operated  under  such  various  condi- 
tions as  may  be  necessary  to  determine  the  losses.      By 
loading  the   alternator   and    simultaneously  measuring 
its  output  and  the  power  absorbed  by  the  motor,  the 
efficiency  of  the  alternator  may  be  determined  with  suffi- 
cient accuracy  for  ordinary  purposes.      In  each  case  the 
power  supplied  to  the  alternator  is  equal  to  the  power 
absorbed  by  the  motor  multiplied  by  the  efficiency  of 
the   motor   given    in    per   cent.     The    motor    may  be 
rated  by  determining  its  efficiency  by  the  stray  power 
method.     If  the  motor   efficiency  for  various   loads  is 
determined  by  some  exact  method  (Vol.  I.,  p.  255),  the 
power  transmitted  to  the  alternator  may  be  determined 
with    considerable    exactness.     When    it    is   desired   to 
carry  the  accuracy  of  this  method  of  testing  beyond  a 
fairly   good    approximation,   the    efficiency   at    various 
loads  of  the  rated  motor  should  be  plotted,  so  that  the 
efficiency  at  any  load  may  be  readily  read  off. 

3.  Mordey  s  MetJiod.    A  neat  arrangement  for  testing 
a  single  alternator  by  a  method   akin   to   Hopkinson's 
method  has  been  devised  by  Mr.  Mordey.*     It  is  to  be 
remembered  that  the  Mordey  alternator  has  a  stationary 
armature,   the    individual   coils   of   which   may  be   con- 
nected in  any  desired  combination  with  perfect  facility. 

*  Testing  and  Working  Alternators,  Jour.  Inst.  £.  £.,  Vol.  22,  p.  116. 


376  ALTERNATING   CURRENTS. 

Such  an  armature  may  be  divided  into  two  parts  which 
are  connected  in  such  a  way  as  to  oppose  each  other 
(Fig.  170).  If  one  part  gives  a  somewhat  higher  press- 
ure than  the  other,  the  first  part  will  operate  as  a  gen- 
erator and  the  second  as  a  motor  if  the  alternator  is 
driven  in  the  usual  manner.  By  properly  adjusting  the 
difference  between  the  pressures  of  the  two  parts,  the 
current  flowing  in  the  machine  may  be  caused  to  have 


Pig.  17O  a 

any  desired  value.  The  efficiency  of  the  machine  is 
gained  by  measuring  the  power  absorbed  by  the  ma- 
chine operating  as  a  self-contained  motor-generator  and 
measuring  its  output  by  a  wattmeter ;  the  pressure  coil 
of  the  wattmeter  being  connected  across  the  terminals 
of  the  motor  and  generator  coils,  and  the  series  coil 
being  connected  directly  in  the  circuit.  The  power 
thus  measured  by  the  wattmeter  is  evidently  about  one- 
half  of  the  total  energy  of  the  machine;  consequently 
the  corrected  efficiency  is 


EFFICIENCIES. 
2    W 


377 


where  P  is  the  power  supplied  to  the  machine.  The 
difference  in  the  pressure  developed  in  the  parts  of 
the  machine  which  is  necessary  to  cause  the  desired 
current  to  circulate  may  be  caused  by  an  unsymmet- 
rical  division  of  the  armature  (Fig.  1700),  or  by  supply- 


Fig.  17O  b 

ing  a  little  additional  pressure  to  the  generator  side  by 
means  of  a  transformer.  The  latter  may  be  supplied 
from  another  alternator  operating  in  synchronism  with 
the  machine  under  test,  or  it  may  be  supplied  directly 
from  the  test  machine  (Fig,  170  b).  The  transformer  is 
likely,  however,  to  introduce  uncertain  elements  of  loss. 
4.  Ayrton's  Method.  As  pointed  out  by  Professor 
Ayrton,*  this  arrangement  may  be  modified  so  as  to 
apply  to  alternators  with  rotating  armatures.  In  this 
case  opposite  halves  of  the  fields  are  magnetized  in  oppo- 

*  your.  Inst.  E.  E.t  Vol.  22,  p.  136. 


373 


ALTERNATING   CURRENTS. 


site  directions,  and  the  armature  is  short-circuited 
through  an  amperemeter  (Fig.  171).  By  adjusting  the 
relative  excitation  of  the  halves  of  the  fields  a  current 
of  any  desired  value  may  be  caused  to  circulate  in  the 
armature.  If  the  excitation  is  practically  normal,  the 
measured  losses  of  the  machine  when  any  current  is 
flowing  will  be  practically  equal  to  those  when  the  ma- 


Pig.  171 

chine  is  operating  normally  on  the  same  current,  pro- 
vided we  may  assume  that  the  losses  in  an  alternator  are 
appreciably  equal  when  driven  as  a  generator  and  as  a 
motor.  This  assumption  seems  entirely  reasonable. 
The  arrangement  here  described  is  not  applicable  to 
machines  with  armatures  with  the  halves  wound  in 
parallel,  since  under  the  test  conditions  an  amperemeter 


EFFICIENCIES,   ETC. 


379 


connected  between  the  terminals  of  such  an  armature 
would  not  indicate  the  current  circulating  in  the  arma- 
ture coils. 

5.  Motor-Generator  Method.  —  Mordey  has  also  sug- 
gested the  following  purely  electrical  method  of  testing 
alternators  which  have  stationary  armatures.*  The 
machine  being  properly  excited,  one-half  of  the  arma- 
ture is  connected  as  a  generator  to  an  external  load,  R 
(Fig.  172).  The  other  half  of  the  armature  is  con- 


.  172 


nected  to  another  alternator  and  driven  as  a  synchro- 
nous motor.  The  total  losses  under  these  conditions 
are  evidently  equal  to  the  power  absorbed  by  the  motor 
half  of  the  armature  plus  the  exciting  energy  and  minus 
the  output  of  the  generator  half  of  the  armature  ;  that 


is,  Wm  +  C^Rf  -  Wg,  where  Wm  and  Wg  are  respec- 
tively the  power  absorbed  by  the  motor  and  the  out- 
put of  the  generator.  Since  the  output,  Wg,  is  the 


*  Jour.  Inst.  E.  £.,  Vol.  22,  p.  122. 


380 


ALTERNATING   CURRENTS. 


output  of  only  half  the  armature,  but  the  losses  thus 
determined  are  those  of  the  whole  machine,  the  losses 
will  be  the  same  when  the  machine  is  run  as  a  gen- 
erator with  an  output  of  2  Wg,  provided  generator 
losses  and  motor  losses  are  the  same.  The  efficiency 
of  the  machine  as  a  generator  is  therefore 

= 2   Wg 2   Wg 

2  Wg  ¥  Wm  +  C*Rf  -  Wg      Wg  +  Wm  +  C*R'f 

6.    By  Driving  as  a  Synchronous  Motor —  Applicable  to 
Machines  with  either  Revolving  Armatures  or  Fields. — 


Fig.  173 


Another  somewhat  similar  plan  is  to  divide  the  station- 
ary armature  into  three  divisions  which  are  connected  in 
series.  Two  of  these  are  made  equal  and  are  connected 
in  opposition.  The  third  consists  of  only  a  small  portion 
of  the  armature.  The  machine  is  electrically  connected 
to  another  alternator  and  operated  as  a  synchronous 
motor  under  the  influence  of  the  small  armature  divis- 
ion (Fig.  173).  By  a  proper  choice  of  the  number  of 
coils  composing  the  small  division  any  desired  current 


EFFICIENCIES,   ETC. 


381 


may  be  sent  through  the  machine  to  be  tested.  The 
energy  given  to  the  machine  represents  the  losses  in 
the  machine  when  operated  as  a  generator  and  produc- 
ing the  same  current  with  the  same  excitation.  This 
method  is  also  applicable  to  determining  the  losses  in 
machines  with  revolving  armatures  the  coils  of  which 
are  all  connected  in  series.  In  this  case  the  fields  are 


EXCITATION 


Fig.  174 

excited  with  the  poles  on  one-half  reversed  (Fig.  174), 
one-half  of  the  field  being  slightly  stronger  than  the 
other.  If  the  armature  is  connected  to  that  of  another 
alternator,  it  will  run  as  a  motor.  The  instantaneous 
counter  electric  pressure  of  the  machine  under  test 
depends  upon  the  relative  strength  of  the  two  halves 
of  the  field,  and  by  adjusting  this,  with  due  reference 


382  ALTERNATING   CURRENTS. 

to  the  impressed  pressure  which  should  be  of  a  relatively 
small  value,  the  current  flowing  in  the  armature  circuit 
may  be  given  any  desired  value.  Under  these  condi- 
tions the  losses  in  the  test  machine  are  equal  to  the 
power  absorbed. 

99.  Shop  Tests.  —  When,  in  either  of  the  cases  men- 
tioned heretofore,  the    operation  of  an  alternator  as  a 
motor  is  predicated,  it  is  assumed  either  that  the  test 
machine  is  brought  to  synchronism  with  the  alternating 
source,  or  that  the  primary  generator  is  started  from  a 
state  of  rest  in  which  case  the  test  machine  will  start 
and  run  with  it. 

These  methods  of  testing  are  not  only  convenient  in 
determining  the  losses  and  the  efficiency  of  an  alternator, 
but  the  tests,  according  to  several  of  the  methods,  are 
made  with  the  consumption  of  comparatively  little  power. 
This  makes  the  methods  satisfactory  for  use  in  shop  tests 
for  determining  the  reliability  in  operation  and  the  heat- 
ing limits  of  machines.  Mordey  has  suggested  that  the 
efficiency  of  an  alternator  with  stationary  armature  may 
be  determined  from  a  test  of  one  armature  coil,  but  this 
is  only  approximate  and  cannot  serve  as  a  satisfactory 
shop  test  which  requires  a  test  of  the  complete  machine. 

100.  Wattmeters   on    High    Pressure    Circuits.  —  In 
using  wattmeters  where  high   pressures   are  met,  con- 
siderable difficulty  is  found  in  arranging  a  satisfactory 
non-inductive   resistance   for   the   pressure  coil.      This 
difficulty  may  be  readily  overcome  by  a  plan  suggested 
by  Dr.  J.  A.  Fleming,*  and  which  was  successfully  used 
in    testing   high-pressure   alternators    at    the    Chicago 

*  Jour.  Inst.  E.  E.,  Vol.  22,  p.  99;   London  Electrician,  Vol.  30,  p.  455. 


EFFICIENCIES,   ETC. 


383 


WATTMETER 


SHUNT 


World's  Fair.  Instead  of  connecting  the  pressure  coil 
of  the  wattmeter  across  the  terminals  of  the  test  circuit, 
the  primary  of  a  transformer 
is  so  connected,  and  the  press- 
ure coil  of  the  wattmeter  is 
connected  to  the  secondary 
of  the  transformer  (Fig.  175). 
The  constant  of  the  watt- 
meter is  then  dependent  upon 
the  ratio  of  transformation 
of  the  transformer,  which  may  £ 
be  readily  measured.  This 
method  gives  entirely  reliable 
results,  since  the  phases  of 
the  primary  and  secondary 
pressures  of  a  very  lightly 
loaded  transformer  are  almost 
exactly  180°  apart.  If  the 
wattmeter  constant  be  deter- 
mined without  the  trans- 
former, its  constant  when  in 
use  with  the  transformer  must  be  multiplied  by  the 
ratio  of  transformation. 

101.  Variation  of  Efficiency,  Weight,  and  Cost,  with 
Output.  —  In  general,  it  is  safe  to  say  that  the  efficiency 
of  an  alternator  should  be  very  nearly  the  same  as  that 
of  a  continuous-current  machine  of  the  same  size  and 
built  for  the  same  duty.  The  electrical  losses  in  the 
two  types  of  dynamos  should  differ  but  little,  and  the 
conversion  losses  can  be  held  to  an  appreciable  equality 
by  using  proper  magnetic  densities  in  alternators,  and  by 


2400 


Fig.  175 


ALTERNATING   CURRENTS. 


proper  construction.  The  curve  showing  the  relation 
of  commercial  efficiency  to  output,  given  in  Fig.  127  of 
Vol.  I.  (reproduced  in  Fig.  176),  may  therefore  be  taken 
to  fairly  represent  alternator  efficiencies.  For  capaci- 
ties greater  than  100  K.W.,  the  efficiency  increases  at 
a  very  slow  rate  towards  a  limit  of  about  95  per  cent. 


95 


90 


85 


81 


50 
OUTPUT. 

Fig.  176 


75 


100 


The  great  Westinghouse  alternators  of  1000  horse  power 
capacity,  which  were  tested  at  the  World's  Fair  at  Chi- 
cago, showed  a  commercial  efficiency  of  upwards  of  96 
per  cent.  The  reported  efficiency  of  various  alternators 
of  a  capacity  of  about  200  K.W.,  varies  between  93  per 
cent  and  95  per  cent.  The  efficiencies  of  continuous.- 
current  machines  of  great  capacity  are  shown  by  test 
to  be  about  the  same. 


EFFICIENCIES,    ETC. 


385 


The  economic  curve  for  alternators  is  quite  similar  in 
form  to  those  given  for  continuous-current  machines 
(Vol.  L,  p.  268).  Figure  177  gives  the  experimentally 
determined  curve  for  an  alternator  of  several  hundred 
kilowatts  output. 


H  M  K 

PROPORTION     OF    FULL     LOAD. 

Figr.  177 

The  relation  between  weight  and  output,  which  has 
already  been  discussed  in  reference  to  continuous- 
current  dynamos  (Vol.  L,  p.  262),  holds  equally  for 
alternators,  as  does  also  the  relation  between  output 
and  cost.  Well-designed  continuous-current  machines 
of  a  greater  capacity  than  10  K.W.  weigh  between  75 
and  200  pounds  per  kilowatt,  depending  upon  the  type 
of  machine,  the  material  of  which  the  magnetic  circuit 


386 


ALTERNATING   CURRENTS. 


is  composed,  the  capacity  of  the  machine,  and  the  use 
for  which  it  is  designed.  Practically  the  same  may  be 
said  of  well-designed  single-phase  alternators.  The 
almost  universal  use  of  toothed  cores  in  alternators  of 
the  latest  design  has  decreased  their  weight  as  com- 
pared with  continuous-current  machines  having  smooth 
cores.  But  the  use  of  toothed  cores  in  the  armatures 


Fig.  178  a 


of   continuous-current    machines    makes    an    equal    im- 
provement in  them. 

If  the  entire  surface  of  alternator  armatures  could  be 
effectively  covered  with  wire,  as  in  continuous-current 
machines,  the  high  allowable  periphery  speeds  and  other 
conditions  already  discussed  (compare  Sect.  6)  would 
give  them  a  largely  increased  output  per  unit  of  weight. 
It  has  already  been  shown  that  this  cannot  be  done  for 
machines  with  armatures  of  one  circuit.  It  is  possible, 
however,  to  wind  the  armatures  with  two  circuits,  which 


EFFICIENCIES,   ETC.  387 

together  entirely  fill  the  surface,  each  circuit  occupying 
one-half  the  winding  space,  as  in  single-circuit  machines, 
the  coils  being  arranged  alternately.  The  pressure 
waves  of  two  circuits  thus  arranged  have  a  phase  differ- 
ence of  90°  (Fig.  179).  It  is  also  possible  to  economically 
fill  the  winding  space  with  three  sets  of  independent 
and  overlapping  coils,  which  are  connected  in  three  in- 
dependent circuits  (Fig.  178).  In  this  case  the  press- 
ure waves  of  the  different  circuits  follow  each  other 
with  a  phase  difference  of  120°  (Fig.  180). 

Alternators  which  produce  a  single  pressure  wave 
are  called  Single-phase  Machines,  or  Single-phasers. 
Those  that  produce  more  than  one  pressure  wave  are 
called  Poly-  or  Multi-phase  Machines,  or  Poly-  or  Multi- 
phasers.  These  are  divided,  according  to  the  number 
of  circuits,  into  Two-phasers,  Three-phasers,  etc. 

The  conditions  at  present  existing  in  the  construc- 
tion of  polyphase  machines  enable  them  to  be  built  at 
a  considerable  reduction  of  cost  and  weight  per  unit  of 
output  as  compared  with  single-phasers  and  continuous- 
current  machines.  There  is,  however,  a  marked  ten- 
dency toward  a  reduction  in  the  speeds  and  frequencies 
of  such  machines,  and  towards  increase  in  the  solidity 
of  their  construction.  The  result  of  this  is  to  increase 
the  comparative  cost  of  the  machines  per  unit  of  output 
until  they  are  approximately  on  a  level  with  the  best 
continuous-current  dynamos. 

102.  Armature  Reactions  of  Poly-phasers.  — •  The 
armature  reactions  in  poly-phase  generators  is  materi- 
ally different  from  that  in  single-phase  generators. 
Thus,  referring  to  Fig.  89,  it  is  remembered  that  when 


388 


ALTERNATING   CURRENTS. 


CABCABCABCAB 


Pig.  178  b 


EFFICIENCIES,   ETC. 


389 


k    L 


CABr  ABCABCABC 


Fig-.   178  c 


390 


ALTERNATING    CURRENTS, 


the  current  of  a  single-phaser  is  in  phase  with  the 
pressure,  magnetism  is  crowded  into  the  trailing  pole 
tips  at  each  time  of  maximum  current,  and  resumes 
its  initial  position  when  the  current  falls  to  zero.  In 
the  case  of  poly-phase  machines,  in  which  the  wire, 
in  effect,  covers  the  entire  surface  of  the  armature, 
there  is  a  sheet  of  current  at  all  times  under  the  faces 
of  the  pole  pieces,  and  as  this  sheet  of  current  sets 
up  a  nearly  constant  magnetizing  force,  the  skewing 


of  the  magnetic  field,  due  to  armature  cross-turns,  is 
practically  constant.  The  effect  of  back-turns  in  a 
single-phase  machine  having  a  ring  armature  is  repre- 
sented in  Fig.  iSia,  two  coils,  a,  a,  being  shown.  The 
ordinates  of  the  curve  d  represent  the  magnetizing 
effect  of  the  armature  coils  which  aids  or  opposes  the 
field  magnetization  when  the  centres  of  the  coils  are 
in  different  positions  corresponding  to  the  ordinates, 
and  a  current  of  constant  strength  is  passed  through 
them.  This  magnetizing  effect  or  activity  must  evi- 


EFFICIENCIES,    ETC. 


391 


dently  be  zero  when  the  centres  of  the  coils  are 
directly  under  the  poles.  E  is  the  pressure  curve  of 
the  machine,  and  c  the  current  curve  in  phase  with 
the  pressure.  By  combining  the  current  curve  with  the 
curve  of  activity,  the  actual  effect  of  the  alternating 
current  in  the  armature  coils  may  be  obtained,  as  is 
shown  by  curve  k.  It  is  seen  that  the  sum  of  the  posi- 


Pig.  ISO 

tive  and  negative  effects  is  zero,  but  that  they  have  a 
periodical  disturbing  effect  upon  the  field. 

In  the  same  manner  it  is  shown  in  Fig.  181^,  that 
if  the  current  lags  behind  or  leads  the  pressure,  one 
loop  of  the  curve  k1  is  greater,  and  the  fields  are 
periodically  either  weakened  or  strengthened. 

In  the  case  of  a  two-phase  alternator,  a  second  curve 
of  reaction  exactly  similar  to  k  and  one-fourth  the  pitch, 
or  90°,  from  k,  may  be  drawn  like  the  heavy  lines  in 
Fig.  i8i<:  to  represent  the  action  of  the  second  set  of 
coils,  and  the  magnetizing  effect  of  the  two  phases  is 
added  together  and  the  skewing  effect  becomes  nearly 


392 


ALTERNATING    CURRENTS. 


uniform  when  the  armature  current  and  pressure  are  in 
phase.  Similarly,  when  the  current  lags  or  leads,  there 
is  a  constant  weakening  or  strengthening  of  the  fields, 
as  shown  in  Fig.  i8i£,  where  the  ordinates  of  the  line 
k1'  represent  this  effect.  In  case  the  current  leads,  the 


Fig.  181 

fields  will  be  strengthened.  By  the  same  process  it 
may  be  shown  that  the  armature  reactions*  in  any  poly- 
phase machine  is  practically  constant  and  is  greater  in 

*  Ossian  Chrytneus,  Armature   Reactions,   Electrical    World,  Vol.  25, 
p.  450. 


EFFICIENCIES,   ETC.  393 

its  effect  than  in  a  single-phase  machine.  In  machines 
having  large  armature  reactions  this  may  limit  the 
output  of  poly-phasers,  but  in  those  with  small  arma- 
ture reactions  poly-phase  armatures  have  much  greater 
capacity  than  single-phasers. 

102  #.  Connecting  up  Poly-phase  Armatures.  —  The 
connecting  up  of  two-phase  armatures  is  very  simple. 
If  the  armature  is  wound  with  independent  concentrated 
coils,  each  coil  may  be  connected  to  its  individual  col- 
lector rings,  in  which  case  four  rings  are  required  and 
the  two  circuits  are  entirely  independent ;  or,  one  ter- 
minal of  each  coil  may  be  connected  to  a  common  col- 
lector ring  and  the  other  terminals  to  independent  rings, 
in  which  case  but  three  rings  are  required  and  the  cir- 
cuits have  a  common  point.  If  the  armature  has  a  con- 
tinuous-current or  closed-circuit  distributed  winding, 
such  as  is  treated  in  Section  5,  it  may  be  made  into  a 
two-phaser  by  connecting  collector  rings  to  the  wind- 
ings at  the  ends  of  two  diameters  which  are  90°  apart, 
if  the  machine  is  bipolar ;  if  the  machine  is  multipolar, 
the  connections  must  be  made  as  described  in  Section 
191.  Four  rings  are  necessary  in  this  case,  as  the  use 
of  a  common  ring  would  cause  the  permanent  short- 
circuiting  of  one-quarter  of  the  armature.  A  continu- 
ous-current armature  converted  into  a  polyphaser  (of 
any  number  of  phases)  in  this  manner  has  a  capacity 
when  used  as  a  poly-phase  alternator  which  is  equal  to 
its  continuous-current  capacity,  though  it  has  already 
been  shown  that  its  capacity  as  a  single-phaser  is  only 
seven-tenths  as  great  as  its  continuous-current  capacity 
(Sect.  5). 


394  ALTERNATING   CURRENTS. 

The  manner  of  connecting  three-phase  armatures  is 
not  so  immediately  evident,  but  is  perfectly  simple.  It 
is  illustrated  in  Figs.  178$  and  178^.  Three  collect- 
ing rings  are  universally  used,  and  if  the  armature  is 
wound  with  three  independent  coils,  these  may  be  con- 
nected to  the  rings  in  either  of  two  ways :  (i)  one  end  of 
each  of  the  coils  may  go  to  a  common  point,  and  the 
other  ends  go  to  independent  rings ;  or,  (2)  the  coil 
terminals  may  be  connected  together  two  and  two, 
forming  a  sort  of  triangle,  and  connections  be  carried  to 
the  collector  rings  from  these  points.  The  latter  ar- 
rangement makes  each  coil  terminate  at  both  ends  in 
a  collector  ring,  and,  since  there  are  six  coil  ends  and 
three  rings,  each  ring  is  connected  with  two  coil  termi- 
nals. These  two  arrangements  are  illustrated  respec- 
tively in  Figs.  178^  and  178^,  each  of  which  shows  the 
winding  diagrammatically  as  developed  and  as  pro- 
jected. The  following  considerations  make  it  perfectly 
easy  to  connect  the  coils  in  the  proper  order:  Fig.  180 
shows  that  when  the  current  in  one  coil  is  at  its  maxi- 
mum point,  the  currents  in  the  other  two  are  equal  to 
each  other  and  opposite  to  the  direction  of.  the  first; 
then,  considering  the  instant  at  which  the  conductors  of 
one  coil  (such  as  C  in  the  figure)  are  directly  under  the 
poles,  if  we  connect  its  positive  end  to  the  common  or 
Neutral  Point,  the  negative  ends  of  the  other  two  coils 
must  be  connected  to  the  same  point.  Each  of  the  free 
terminals  may  then  be  connected  to  one  of  the  collector 
rings,  and  the  connection  is  completed  according  to  the 
first  or  Star  arrangement  (Fig.  178  b).  To  make  the  sec- 
ond or  Mesh  arrangement,  the  coils  must  be  connected 


EFFICIENCIES,   ETC.  395 

so  that,  at  the  instant  considered,  the  current  flows  by 
two  paths  through  the  armature  from  the  negative  to  the 
positive  terminal  of  the  first  coil  (coil  C  of  the  figure). 
Consequently  the  negative  terminals  of  the  first  and  sec- 
ond coils  go  to  one  collector  ring,  the  positive  terminals 
of  the  first  and  third  coils  to  another  ring,  and  the  free 
terminals  of  the  second  and  third  coils  to  the  third  ring 
(Fig.  178*). 

If  the  armature  has  a  closed-circuit  or  distributed 
winding,  the  connection  is  very  simple,  as  the  rings  are 
connected  to  points  in  the  windings  120°  apart.  If  the 
machine  is  multipolar,  it  must  be  remembered  that  one 
cycle,  or  360  electrical  degrees,  is  comprised  within  the 
space  of  twice  the  polar  pitch.  This  is  more  fully 
treated  in  Section  191. 

In  a  three-phase  machine,  if  the  armature  is  mesh- 
connected,  the  pressure  between  any  two  collector  rings 
is  equal  to  the  pressure  developed  in  one  coil,  while  the 
current  leaving  a  brush  is  the  vector  sum  of  the  current 
in  two  coils ;  and,  if  the  armature  is  star  connected,  the 
pressure  between  the  rings  is  equal  to  the  vector  sum 
of  the  pressure  developed  in  two  coils,  while  the  current 
leaving  a  brush  is  equal  to  the  current  in  a  coil.  The 
vector  sum  in  either  case  is  A/3  times  the  arithmetical 
sum,  so  that  the  capacity  of  a  machine  is  independent 
of  the  way  in  which  its  armature  is  connected,  but,  for 
a  given  pressure  and  output,  the  windings  will  differ 
(though  the  weight  of  copper  will  be  the  same)  for 
the  two  arrangements.  This  is  treated  more  fully  in 
Chap.  XIII. 


396  ALTERNATING   CURRENTS. 


CHAPTER  IX. 

MUTUAL    INDUCTION. 

103.  The  Function  of  a  Transformer.  —  The  remarka- 
ble development  in  the  use  of  alternating  currents  for 
transmitting  and  distributing  electric  power,  is  mainly 
due  to  the  facility  and  economy  with  which  they  may 
be  transformed  from  one  pressure  to  another.  The 
transmission  of  electric  power  between  two  points  may 
be  made  by  alternating  currents  at  high  pressure  for 
the  sake  of  economy  in  the  cost  of  conductors,  and 
the  pressure  may  be  reduced  at  the  receiving  point 
by  means  of  induction  coils  to  any  value  which  is 
deemed  entirely  safe  and  convenient  for  distribution. 

The  induction  coils  that  are  used  for  this  purpose 
are  called  Transformers  or  Converters,  because  they 
are  used  to  transform  or  convert  the  electrical  energy 
from  one  state  to  another.  Formerly  transformers  were 
sometimes  called  Secondary  Generators  on  account  of 
the  apparent  regeneration  of  the  energy  of  the  alternat- 
ing currents. 

The  action  of  transformers  is  due  to  the  inductive 
effect  which  a  varying  current  in  one  circuit  exerts 
upon  an  adjacent  circuit.  Since  this  effect  is  a  mut- 
ually interacting  one,  it  is  called  Mutual  Induction, 
Before  proceeding  to  the  study  of  the  commercial  trans- 


MUTUAL   INDUCTION.  397 

formers,  it  is  essential  to  examine  the  relations  which 
exist  between  two  adjacent  circuits,  one  or  both  of 
which  carry  an  alternating  current. 

104.  Mutual-Inductance.  —  Suppose  two  adjacent  coils 
surrounded  by  air  and  in  which  a  current  is  flowing  : 
the  total  number  of  lines  of  force  passing  through 
either  one  of  the  coils  is  the  number  of  lines  set  up 
by  the  current  in  that  coil  plus  the  number  of  lines 
set  up  in  the  other  coil  which  are  embraced  by  the 
first.  If  the  current  is  changed  in  either  of  the  coils, 
an  electrical  pressure  is  set  up  in  the  coil  under  con- 
sideration, which  may  be  called  the  first  coil,  which  is 

equal  to  E'  —  ^—  —  ,  where  n'  is  the  number  of  turns 

i&di 

and  N'  the  magnetic  flux  in  the  first  coil.  The  num- 
ber of  lines  passing  through  the  coil  due  to  its  own 

icfi  I  '  C1 
current  is  --  -  -  .     Ll    and    C'    are   respectively   the 

self-inductance  and  the  current  in  the  first  coil  (Chap. 
III.).  The  number  of  lines  of  force  due  to  the  first  coil 
which  pass  through  the  second  coil  evidently  depends 
upon  the  relative  positions  of  the  coils,  but  it  cannot 
be  greater  than  the  total  number  of  lines  set  up  by 
the  current  in  the  first  coil.  For  any  two  fixed  coils 
in  a  medium  of  fixed  permeability,  this  number  of  lines 
is  proportional  to  the  current.  The  electric  pressure 
developed  in  the  second  coil  due  to  the  change  in  the 

nn  dN" 
current  flowing  in  the  first  is  En  =  -  ,  where  n"  and 


N"  are  the  turns  and  magnetism  in  the  second  coil.    The 

rr      n'dN'  ,          ...       ,,,      UdC       , 

equation  E  =  —  -  —  may  be  written  E'  =  -  ,  when 

dt 


398  ALTERNATING   CURRENTS. 

nndNn 
the  permeability  is  constant.    The  equation  En  = 


may  be  similarly  written  E"  =  -  ,   where    io9  M  is 

the  number  of  turns  in  the  second  coil  multiplied  by 
the  number  of  lines  of  force  passing  through  the  second 
coil  which  are  due  to  the  first  coil  when  one  ampere  is 
flowing  in  it  ;  and  M  is  called,  by  analogy,  the  Mutual- 
Inductance  or  the  Coefficient  of  Mutual  Induction  of  the 
coils.  The  value  of  N"  is  evidently  equal  to  N'k,  where 
k  is  a  constant  depending  on  the  reluctance  of  the  path 
of  the  lines  of  force  which  interlink  the  two  coils.  If 

the  coils  are  long  solenoids,  N1  =  £—  -  ,  where  A  is 

io/ 

the  cross-section  and  /  the  length  of  the  coil  ;  and  if  the 
solenoids  are  wound  one  over  the  other  so  that  their 
dimensions  are  practically  equal,  the  value  of  k  is  unity 
because  N"  evidently  becomes  equal  to  N!  ,  and  Mbc- 

N'nn 

comes  equal  to  —  —  —  •     If  the  action  of  the  coils  be  now 
icPC 

reversed,  that  is,  if  a  current  flows  in  the  second  coil,  we 

,„       4Trn"CA        ,   n/r       N\n'      B  ,    Nr       »' 

have  Wi  =  *  -  ;  —  ,  and  Ml  =  —  Jr~*     But  —  -  =  — 

io  /  io8C  N\      n" 

and  hence  M=  M-,.     In  this  case,  also,  L'  =  —  T-—  and 

io8  C 


L"  =       in.    Consequently  L'L"  =  M*,  or  M=  ^/LrLrr. 

io8  C 
If  the  solenoids  be  now  separated  by  drawing  one 

out  of  the  other,  the  value  of  M  continually  decreases 
as  the  separation  continues.  The  self-inductances  of 
the  coils  remain  constant,  so  that  as  the  coils  sepa- 
rate, the  mutual-inductance  becomes  less  than  -VL'L". 
As  the  separation  of  the  coils  becomes  greater,  the 
value  of  M  decreases  towards  a  minimum  of  zero.  It 


MUTUAL   INDUCTION.  399 

reaches  this  value  when  the  coils  are  at  an  indefinitely 
great  distance  apart  or  the  axis  of  one  is  placed  symmet- 
rically but  at  right  angles  with  reference  to  the  axis 
of  the  other.  The  maximum  possible  value  of  the 
mutual-inductance  of  the  two  coils  is  therefore  a  mean 
proportional  between  the  values  of  their  self-induc- 
tances, and  the  minimum  value  is  zero,  The  maximum 
value  can  only  be  attained  when  all  the  lines  of  force 
due  to  one  coil  pass  through  all  the  turns  of  the 
other,  and  the  value  of  M  may  therefore  be  written 
M<'VL'L")  from  which  it  is  at  once  seen  that  the 
mutual-inductance  of  the  two  coils  must  always  be 
small  if  the  self-inductance  of  one  or  both  of  the  coils  is 
very  small,  while  the  mutual-inductance  may  be  large  if 
both  the  self-inductances  are  large.  It  is  shown  on  the 
preceding  page  that  the  value  of  M  may  be  generally 
written  M=nnN'k.  In  this  expression  N'  is  propor- 
tional to  n',  whence  it  is  shown  that  Mvzn'nnk. 

The  product  of  the  number  of  lines  of  force  which 
interlink  two  coils  by  the  number  of  turns  in  the  indi- 
vidual coils  will  not  change  by  changing  the  point  of  ref- 
erence from  one  coil  to  the  other,  and,  consequently,  the 
mutual-inductance  of  two  coils  is  always  the  same  when 
measured  from  either  coil,  provided  the  reluctance  of  the 
magnetic  circuit  is  unchanged.  When  the  magnetic 
circuit  is  composed  wholly  or  partly  of  iron,  it  is  neces- 
sary to  have  an  equal  number  of  lines  of  force  in  each 
part  of  the  circuit,  when  the  two  measurements  are 
made,  in  order  that  they  may  give  equal  results.  When 
the  magnetic  circuit  is  made  up  partly  of  iron  and  partly 
of  non-magnetic  materials,  and  the  coils  are  quite  dif- 


400  ALTERNATING   CURRENTS. 

ferent,  as  in  dynamos,  the  condition  of  equal  induction 
is  difficult  to  fulfil,  and  the  value  of  M  may  be  quite 
different  when  measured  from  the  fields  and  from  the 
armature.  This  difference  is  wholly  due  to  the  differ- 
ence in  the  permeability  of  the  magnetic  circuit  during 
the  two  measurements. 

As  shown  above,,  the  mutual-inductance  is  homo- 
geneous with,  and  therefore  of,  the  same  dimensions 
as  self-inductance.  Its  unit  is  therefore  the  henry. 
The  practical  unit  for  mutual-inductance,  M,  as  here 
developed  by  comparison  with  L,  is  io9  times  as  large 
as  the  absolute  unit. 

If  the  lines  of  force  due  to  one  coil  which  enter 
another  do  not  all  pass  completely  through  the  sec- 
ond coil,  the  definition  of  the  mutual-inductance  still 
holds  as  already  given,  but  the  summation  of  the  num- 
ber of  lines  of  force  passing  through  each  individual 
turn  must  be  taken  (compare  Sect.  16).  If  iron  be  in- 
serted in  the  path  of  the  lines  which  interlink  two  coils, 

p 
the  mutual-inductance  is  increased  in  the  ratio  of  — •, 

where  P  and  P1  are  the  reluctances  of  the  path  before 

p 

and  after  the  iron  is  inserted.  The  ratio  —  is  depend- 
ent on  the  permeability  of  the  iron  in  the  magnetic 
circuit,  and  the  value  of  M  must  therefore  vary  with 
the  current  in  the  coils  in  any  case  where  iron  is  in 
the  magnetic  circuit,  while  it  is  independent  of  the 
value  of  the  current  when  magnetic  material  is  absent. 

105.  The  Energy  of  Mutual  Induction.  —  Assume  two 
adjacent  coils  with  a  constant  mutual-inductance.  In 
one  let  a  current  of  C'  amperes  flow,  and  in  the  other 


MUTUAL   INDUCTION.  401 

a  current  of  Crl  amperes.  The  number  of  lines  of  force 
due  to  the  first  coil  which  pass  through  the  second  is 

—  jj  —  ,  and  the  change  in  this  number  when  C1  changes 

io8  MdC'      T,    .  ~,  ,  .    ,    ., 

is  -   —  f-f  —  —     If  the  current  G  be  varied,  the  work  due 

to  mutual  induction  which  is  done  in  the  second  coil  in 
changing  the  magnetic  field  against  the  effect  of  the 
current  C"  is  (by  Vol.  L,  p.  69)  dW=MC"dC'  (com- 
pare Sect.  1  8).  If  the  current  in  the  first  coil  be 
changed  from  zero  to  C'  ,  the  work  done  on  the  second 

coil  is  W=  fC  MC'dC  =  MC'C",  which  is  all  stored 

Jo 

in  the  magnetic  field.  If  the  current  C1  falls  again  to 
zero,  this  work  is  restored  to  the  circuit.  When  M 
varies  with  the  current,  the  work  is  still  MC'C",  but 
M  in  the  expression  must  be  assigned  its  equivalent 
mean  value  between  the  limiting  values  of  the  current 
(compare  Sect.  18).  As  the  current  in  the  first  coil 
rises  to  C  ',  the  total  work  stored  in  the  magnetic  field 
is  evidently  the  sum  of  the  work  due  to  self-  and 

LfC'2 
mutual-induction,  or  --  \-MC'C". 

2 

If  the  current  varies  in  both  the  coils  at  the  same 
time,  .the  following  condition  exists  at  any  instant. 
The  total  pressure  in  either  coil  is  the  resultant  of 
three  elements  —  the  active  pressure  (cR),  the  pressure 

of   self-induction    \-~\   and  the  pressure  of  mutual- 


d(Mc"  +  L'c'} 
*  ' 


induction  and 

V  dt  ) 


_ 

2  D 


402  ALTERNATING   CURRENTS. 

,,       d(Mc'  +  L"c") 
~dt  ' 


where  e'  and  e"  are  the  instantaneous  impressed  press- 
ures in  the  two  coils.     By  transformation  we  have 

(e'  -  c'rf)  dt  =  Mdc"  +  L'dJ, 
(,"  _  W)  dt  =  Mdc?  +  L"dc". 

Multiplying  these  respectively  by  c1  and  cn  and  adding, 
gives 

(c'e1  -f  c"e")  dt  -  (W  +  c"  V)  <#  =  £  V<fc'  +  L"c"dc" 
+  M(c'dc"  +c"dc'). 

The  first  and  second  terms  of  the  left-hand  member 
of  this  equation  represent  respectively  the  total  work 
done  by  the  impressed  electric  pressures  and  the  work 
expended  in  the  coils  in  heat  during  the  interval  dt. 
Their  difference  represents  the  work  done  on  the 
magnetic  field.  The  total  work  done  on  the  mag- 
netic field  during  any  change  of  the  currents,  as  from 
zero  to  C'  and  C"  ,  is  found  by  integrating  the  right- 
hand  member  of  the  equation.  Thus, 

c'dc'  +  L"        c"dc"  +  M°  (c'dc"  +   c"dc>) 


2  2 

If  the  currents  now  fall  to  zero  again,  the  work  has 
the  same  value  as  above,  but  the  negative  sign.  In  the 
first  case  electrical  energy  is  absorbed  from  the  circuit 
and  stored  in  the  magnetic  field  which  is  set  up,  and  in 
the  second  case  the  stored  work  is  restored  to  the  cir- 
cuit as  the  magnetic  field  dies  away. 


MUTUAL   INDUCTION.  403 

106.  Transfer  of  Electricity  by  the  Effect  of  Mutual 
Induction.  —  Now  suppose  that  no  pressure  is  initially 
impressed  on  the  second  coil  (that  is,  e"  =  o),  then 
when  the  current  in  the  first  coil  is  changed,  the  con- 
ditions in  the  second  coil  are  given  from  the  equations 

above  ;  thus 

„_  -  d(Mc<  +  L"c") 

f"dt 
Whence          c"r"Jt  =  -  Mdc'  -  L"dc", 


and  <*'*"  dt  =  ~ 


Since  the  last  term  reduces  to  zero,  the  quantity  of 
electricity  which  is  transferred  in  the  second  coil  under 
the  inductive  influence  of  the  first  when  its  current 
chanes  from  zero  to  O  is 

MC>    MB 


If  the  current  of  the  first  coil  is  now  brought  to  its 
original  value,  we  have 

M  r°,  ,      L"  r  °  ,  „      MO 

-         dc  =  ~ 


The  two  quantities  are  equal  and  of  opposite  sign,  so 
that  the  transfer  of  electricity  in  the  secondary  coil 
during  the  rise  and  fall  of  the  primary  current  reduces 
to  zero,  provided  the  original  and  final  values  of  the 
primary  current  are  equal  (compare  Sect.  19).*  If  the 
current  in  the  first  coil  is  a  simple  periodic  one,  a 

*  Gera.rd's   Lemons  sur   r  Alectridte,   3d  ed.,  Vol.  I.,  p.  227;    Hospi- 
taller's Traite  sur  r£nergie  £lectriquey  Vol.  I.,  p.  486. 


404         ALTERNATING  CURRENTS. 

periodic  current  of  the  same  frequency  is  set  up  in  the 
second  coil.  Such  an  arrangement  of  two  coils  is  a 
transformer.  The  first  coil  is  called  the  Primary  Coil, 
and  the  second  is  called  the  Secondary  Coil.  The 
pressures  or  currents  in  the  primary  and  secondary 
coils  are  called  respectively  primary  and  secondary  press- 
ures or  currents.  When  the  primary  current  wave  is  a 
sinusoid  and  the  mutual-inductance  is  constant,  the 
electric  pressure  induced  in  the  secondary  is  also  a 
sinusoid,  but  lagging  in  phase  90°  behind  the  phase 
of  the  primary  current.  This  is  evident  from  the  fact 
that  the  induced  pressure  is  proportional  to  the  rate 
of  change  of  the  magnetization,  and  the  magnetization 
is  in  phase  with  the  primary  current  (compare  Sect.  15). 
When  iron  is  present  in  the  magnetic  circuit,  M  is  no 
longer  constant,  and  the  rate  of  change  of  the  magnet- 
ization is  not  proportional  to  the  rate  of  change  of  the 
current ;  consequently  the  secondary  pressure  wave  is 
no  longer  similar  to  the  wave  of  primary  current,  but 
it  is  always  exactly  similar  in  form  to  the  wave  of 
counter  electric  pressure  set  up  in  the  primary  coil. 

107.  The  Pressure  Relations  in  a  Transformer.  —  If  a 
sinusoidal  current  is  caused  to  flow  in  one  of  two  coils, 
such  as  have  been  considered  in  the  preceding  para- 
graph, the  relative  positions  of  the  pressures  in  the  two 
coils  maybe  shown  graphically  as  follows.  In  Fig.  182, 
let  OEla  be  the  active  pressure  in  the  primary  coil  act- 
ing upon  the  current  (OC^.  This  current  will  set  up  a 
self-inductive  pressure  in  the  primary,  OElt  =  2  irfL^C^ 
and  a  mutually  inductive  pressure  in  the  secondary, 
OElm  =  2  TrfMC^  These  pressures  are  in  the  same 


MUTUAL   INDUCTION. 


405 


direction  and  lag  90°  behind  the  current  (see  Sect.  106). 
The  pressure  OElm  will  set  up  a  current  (OC2)  in  the 


Figf.  182 


secondary  which  will  cause  a  pressure  in  the  secondary, 
Zs  =  2  7r/Z2(72,  and  a  pressure  in  the  primary,  O£2m 


406  ALTERNATING   CURRENTS. 


=  2  TrfMCfr  both  lagging  90°  behind  the  current.  The 
active  pressure  in  the  secondary  is  the  resultant  of  OElm 
and  OE2s,  or  OE^.  The  pressure  impressed  upon  the 
primary  (OE-^  must  be  such  that  when  combined  with 
the  self  and  mutually  inductive  pressures  OEla  and 
OE2m  the  resultant  will  be  the  active  pressure  OEla. 
The  vector  diagram  which  gives  OEl  is  completed  by 
drawing  from  Ela  the  line  ElaA  which  is  equal  and 
parallel  to  OE2m,  and  from  A  the  line  AE1  which  is 
equal  and  parallel  to  OEls.  Then  OEl  is  the  impressed 
pressure.  If  the  current  OC2  be  increased,  E2s  and  E2m, 
which  depend  upon  the  secondary  current,  will  be  larger, 
and  (/>!,  the  angle  of  lag  of  the  primary  current,  will 
be  less,  while  the  secondary  current  will  swing  around 
more  nearly  into  opposition  with  the  primary  current. 
Under  these  circumstances  the  active  secondary  pressure 
will  be  smaller  if  the  primary  pressure  remains  constant. 
If  the  secondary  current  be  made  smaller,  the  primary 
pressure  remaining  constant,  the  active  secondary  press- 
ure will  be  increased,  ^  increased,  and  the  secondary 
current  will  swing  around  towards  a  phase  which  is  90° 
behind  the  primary  current.  It  is  evident  that  the 
primary  and  secondary  currents  combine  to  give  a 
resultant  magnetizing  effect  which  sets  up  the  magnet- 
ization in  the  magnetic  circuit.  This  property  will  be 
used  in  the  chapter  on  design. 

108.  Measurement  of  Mutual-Inductance.—  Before  leav- 
ing this  part  of  the  subject,  it  is  well  to  consider  the 
methods  of  measuring  mutual-inductances.  The  various 
practical  methods  are  based  on  a  comparison  of  the 
unknown  mutual-inductance  with  a  known  resistance, 


MUTUAL   INDUCTION.  40? 

a  capacity,  a  self-inductance,  or  another  mutual-induc- 
tance. The  latter  may  be  the  mutual-inductance  of  two 
standard  coils  which  are  fixed  in  a  position  relative  to 
each  other.  The  mutual-inductance  of  two  such  coils 
may  be  determined  by  calculation  if  the  coils  are  of 
the  proper  shape,  or  it  may  be  made  by  careful  com- 
parative measurements. 

I.    Direct  Measurement  by  Amperemeter  and  Voltmeter 
(using  an  alternating  current).     The  formula 

cv,  _  MdC' 
~~dT 

(Sect.  104)  indicates  a  method  of  measuring  the  mutual- 
inductance  of  two  coils  when  a  source  of  sinusoidal 
alternating  current  is  at  hand.  When  the  current  is 
sinusoidal  and  flows  continuously  through  the  primary 
coil,  the  instantaneous  pressure  induced  at  any  moment 
in  the  secondary  coil  is 

„  _  Mdc1  •_  Md(c'm  sin  a)  _  Mc'm  cos  ada 
dt  dt  dt 

The  maximum  value  of  the  induced  pressure  is 

..        Me'  mda 
e"    =  =  2 


since         =  27r/  (Sect.    24).      The    effective    value   of 

the   induced   pressure   is   therefore   En  =  2  irfMC'  ,    or 

E" 
M—  --  —  .     The  mutual-inductance  of  the  coils  may 

therefore  be  measured  by  passing  through  one  of  them 
a   sinusoidal   current   the    effective   value  of   which   is 


408 


ALTERNATING  CURRENTS. 


measured  by  an  amperemeter,  and  measuring  the  effec- 
tive value  of  the  induced  pressure  (Fig.  183). 

2.  Direct  Measurement  (using  amperemeter  and  bal- 
listic galvanometer).  Connect  the  primary  of  the  two 
coils,  the  mutual-inductance  of  which  is  to  be  measured, 
in  series  with  a  battery,  an  amperemeter,  and  a  key 
(Fig.  184).  In  series  with  the  secondary  coil  connect  a 
ballistic  galvanometer,  making  the  total  resistance  of  the 
secondary  some  known  value  R" '.  Then,  when  the  key 
in  the  primary  is  closed,  there  will  be  an  induced  current 


in  the  secondary,  during  the  continuance  of  which  the 

MC' 
number  of  coulombs  passing  will  be  Q"  =  (Sect. 

106),  where  C'  is  the  final  value  of  the  current  in  the 
primary,  and  M  is  the  value  of  the  mutual  inductance 
which  is  sought.  The  value  of  Q"  is  determined  from 
the  throw  of  the  ballistic  galvanometer.  The  equation 


MUTUAL   INDUCTION.  409 

then  contains  only  one  unknown  quantity,  the  value  of 
M,  which  is  therefore  determined  by  the  solution 


where  0  is  the  throw  of  the  ballistic  galvanometer,  and 
K  is  its  constant,  If  the  known  primary  current  be 
reversed,  the  formula  becomes 

M=Q"R"  =R"Ke 

'     2C'  2C1    ' 

3.  Comparison  with  a  Known  Capacity  (Carey  Fos- 
ter's Method).  By  modifying  the  preceding  method,  it 
is  possible  to  make  the  desired  determination  without 
knowing  the  constant  of  the  ballistic  galvanometer, 
Thus,  after  the  observations  have  been  taken  as  de- 
scribed, a  condenser  with  the  galvanometer  in  series  may 
be  shunted  around  the  resistance  r1  ,  which  is  in  the 
primary  circuit  (Fig,  185).  Then  when  the  key  is 
closed,  the  quantity  of  electricity  which  passes  through 
the  galvanometer  is  (Sect.  350) 


If  the  resistance  /,  shunted  by  the  condenser,  is  ad- 
justed without  altering  the  total  resistance  in  the 
circuit,  so  that  the  galvanometer  deflections  in  the  two 
positions  are  equal,  or  Ql  =  Q"  ,  then 


whence  M=sr*R". 


ALTERNATING   CURRENTS. 


If  the  deflections,  and  therefore  the  quantities,  of  elec- 
tricity are  not  equal  in  the  two  cases,  this  becomes 

sr'R"0 


where  6  and  0l  are  the  respective  throws  of  the  galva- 
nometer in  the  two  positions.  In  order  that  the  adjust- 
ment may  be  readily  made,  the  arrangement  shown  in 
Fig.  1 86  is  employed.  The  variable  resistances  r1  and 
r",  which  are  in  the  primary  and  secondary  circuits, 


Fig,  185 

respectively,  are  adjusted  until  the  galvanometer  gives 
no  throw  upon  closing  the  primary  circuit.  Then  the 
electric  flow  due  to  charging  the  condenser  is  exactly 
equal  and  opposite  to  the  flow  in  the  secondary.  In 

this  case  we  have 

MC 


sC'r'  = 


R" 


or,  as  before, 


MUTUAL   INDUCTION. 


411 


In  order  that  the  self-induction  of  the  circuits  may 
not  disturb  the  observations,  the  ballistic  galvanometer 
must  have  a  rather  sluggish  needle,  so  that  it  will  not 
move  appreciably  during  the  duration  of  the  discharge  * 
(Sect.  20). 

3  a.  (Pirani's  Method.)  This  method  has  much  in 
common  with  the  preceding,  but  the  arrangement  of  the 
circuits  is  quite  different 
(Fig.  187).  Here  the  pri- 
mary and  secondary  circuits 
each  contain  variable  resist- 
ances, r1  and  r" .  These  are 
connected  together  at  the 
ends  where  they  join  their 
respective  coils  ;  at  the  other 
end  they  are  joined  through 
a  condenser.  The  battery 
and  galvanometer  are  con- 
nected respectively  in  the 
primary  and  secondary  cir- 
cuits, as  shown  in  the  figure. 
When  a  current  is  set  up  in 
the  primary,  a  charging  cur- 
rent tends  to  flow  through  r1'  into  the  condenser  which 
transfers  sC'r'  coulombs  of  electricity  The  average 

difference  of  electric  pressure   at    the  terminals  of  r*' 

sC'^r" 
during  the  period  of  charging,  t,  is  therefore  - 

The  average  pressure  set  up  by  induction  in  the  second- 

*  Philosophical  Magazine,  VoL  23,  3d  Series,  p.  121  ;  Gerard's  Lefons 
sur  r  Electricity  3d  ed.,  Vol.  I.,  p.  327;  Gray's  Absolute  Meas.  in  Elect, 
and  Mag.,  Vol.  II.,  p.  303. 


412  ALTERNATING    CURRENTS. 

ary  circuit,  which  is  opposite  in  direction  to  the  charg- 

MC' 
ing  current,   is  .     When  the  resistances  r1  and  r" 

are  adjusted  so  that  the  galvanometer  shows  no  deflec- 
tion, the  average  fall  of  pressure  in  r"  during  the  period 
of  the  transient  current  which  is  caused  by  the  condenser 
charging  current,  is  equal  to  the  average  pressure  devel- 
oped in  the  secondary  coil.  Consequently, 

sC'r'r"  =  MC't 
t  t 

and  M=sr'r". 

In  this  method,  as  in  the  previous  one,  the  galvanome- 
ter needle  must  be  sufficiently  heavy,  so  that  it  does  not 
move  appreciably  during  the  period  of  the  transient 
current.* 

4.  Comparison  with  a  Known  Self -inductance  by 
Bridge  (Maxwell's  Method).  The  mutual-inductance 
M  of  two  coils  in  this  case  is  compared  with  the 
known  self-inductance  of  one  of  the  coils.  The  coil  of 
known  self-inductance  is  connected  in  one  arm  R  of  a 
bridge,  and  the  other  coil  is  connected  in  the  battery 
circuit  (Fig.  188)  ;  the  connections  being  so  made  that 
the  magnetic  effects  of  the  two  coils  are  in  opposition. 
The  resistances  of  the  other  arms  of  the  bridge  are 
represented  by  R^  A,  and  B.  The  bridge  is  balanced 
by  trial  and  approximation  for  both  steady  and  tran- 
sient currents,  when  the  fall  of  pressure  in  the  bridge 
arm  R  is  equal  to  that  in  the  arm  Rv  From  this  con- 

*  Elektrotechniscke  Zeitschrift,  1887,  Vol.  8,  p.  336;  Hospitaller's  Traite 
de  I'Energie  Electrique,  Vol.  I.,  p.  301. 


MUTUAL   INDUCTION. 


413 


dition  the  following  equations  are  formed.  If  c  and  ^ 
are  the  currents  in  the  arms  R  and  Rv  at  any  instant 
the  fall  of  pressure  in  the  arm  R  is 


dt 


dt 


Fig.  188 


and  the  fall  of  pressure  in  the  arm  R1  is  R^I.  The 
condition  of  balance  for  both  transient  and  steady 
current  requires  that 


r>  r>     ,  Ldc  ,  M(dc  + 

R1c1  =  Re  4-  •  -  H  --  a  —  — 

dt  dt 


, 
and 


„ 

=  Re. 


Hence 


Ldc     M(dc  + 
dt   +"        dt 


That  is,  the  current,  c  +  cv  flowing  in  the  battery  circuit, 
and  that  in  the  arm  R,  c,  must  have  such  a  ratio   that 


ALTERNATING   CURRENTS. 


the  effects  of  self  and  mutual  induction  are  equal  and 
opposite.     Integrating  the  last  equation  gives 

Lc  +  M(c  +  *!>  =  o, 
and  combining  this  with  Re  =  Rfa  gives 


In  order  to  avoid  the  inconvenience  of  the  trial  and 
approximation  method  of  balancing,  a  variable  resist- 


AA/WW 

Fig.  189 


ance,  r,  may  be  connected  between  the  battery  terminals 
of  the  bridge  (Fig.  189),  and  the  required  relations  be- 
tween the  transient  currents  in  the  battery  circuit  and 
arm  R  may  be  gained  by  adjusting  this  resistance  with- 


MUTUAL   INDUCTION.  415 

out  disturbing  the  steady  balance  of  the  bridge.     Then 
we  have,  by  a  solution  similar  to  the  above, 

L        (,  v-R 

TT  —    —       *   ~r  T; 


M  '  *t  '         r       ) 

These  equations  show  that  the  value  of  L  must  be 
greater  than  M,  in  order  that  the  method  may  be  used. 
To  make  the  method  generally  useful,  the  coil  of  known 
self-inductance  should  be  inserted  in  the  shunt  circuit 
with  the  resistance  r.  Then,  when  the  balance  is  made, 

2^ 

M~ 

where  r1  is  the  total  resistance  of  the  shunt  branch. 

In  order  that  this  method  may  be  reliable,  the  induc- 
tances of  the  bridge  coils  must  be  entirely  negligible 
or  a  proper  correction  must  be  made.  To  gain  greater 
sensibility  in  the  method,  a  secohmmeter  may  be  used 
with  the  bridge  (Sect.  37).* 

4  a.  (Niven's  Method.)  In  this  case,  the  mutual-induc- 
tance, M,  of  two  coils  is  compared  with  the  known  self- 
inductance  of  another  coil.  The  coil  of  known  self- 
inductance  is  connected  in  one  of  the  bridge  arms,  R. 
One  of  the  coils  whose  mutual-inductance  it  is  desired 
to  measure  is  connected  in  the  battery  circuit,  and  the 
other  is  connected  in  series  with  a  variable  resistance, 
as  a  shunt  to  the  galvanometer  (Fig.  190).  When  the 
bridge  is  balanced  for  steady  currents,  a  balance  for 


*  Maxwell's  Electricity  and    Magnetism,    2d    ed.,  Vol.    II.,  p.-  365; 
Gray's  Absolute  Measurements^  Vol.  II.,  p.  465. 


416 


ALTERNATING  CURRENTS. 


transient  currents  may  be  gained  by  adjusting  the  vari- 
able resistance  in  the  shunt  circuit.     This  being  done, 

we  have 

L  ^(R-j-K^ 
M  Rr 

where  r  is  the  resistance  of  the  circuit  shunting  the 
galvanometer.     The  galvanometer  needle  must  have  a 


Fig.  190 

considerable  time  of  vibration  as  before,  and  a  secohm- 
meter  must  be  used  to  give  sensitiveness.* 

5.  Comparison  of  Two  Muttial-Inductances  (Maxwell's 
Method).  The  primaries  of  the  two  pairs  of  coils  are 
connected  in  series  with  a  battery  and  key,  and  the 
secondaries  are  connected  in  series  with  variable  re- 
sistances. A  galvanometer  is  connected  as  a  shunt 
between  the  secondaries  (Fig.  191).  The  variable  resist- 


*  Gray's  Absolute  Measurements,  Vol.  II.,  p.  475. 


MUTUAL   INDUCTION. 


417 


ances  are  adjusted  until  the  galvanometer    shows  no 
deflection  upon  opening  and  closing  the  key.     Then 


where  R1  and  R^  are  the  total  resistances  in  the  sec- 
ondary circuit  on  either  side  of  the  galvanometer. 


^-AAA/vVX^AAAAAAAAAAAV^ — . 

(  M.  6s  M.  ) 


Fig.  191 

This  method  may  be  modified  by  connecting  the 
galvanometer  in  series  with  the  secondaries,  which  are 
connected  in  opposition.  A  shunt  is  then  connected 
between  the  lead  wires,  as  in  Fig.  192.  When  the  re- 
sistances on  either  side  of  the  shunt  have  been  adjusted 
so  that  the  galvanometer  shows  no  deflection  on  open- 
ing and  closing  the  key,  the  relation  obtaining  is 


where  Rs  is  the  resistance  of  the  shunt  connection. 
If  the  shunt  connection  is  placed  in  the  primary  circuit. 


4i8 


ALTERNATING  CURRENTS. 


(Fig.  192  a)  and  Rs  is  the  total  resistance  of  the  primary 
circuit  to  the  right  of  the  shunt,  the  relation  becomes 
M»          R, 


M,    R'+R; 


Fig.  192 


Finally  if  a  shunt  is  placed  in  both  primary  and  sec- 
ondary circuits  (Fig.  192  £),  the  relation  becomes* 


Fig-.  192  a 


*  Maxwell's    Electricity  and  Magnetism,    26.    ed.,  Vol.    II.,   p.  363; 
Gray's  Absolute  Measurements,  Vol.  II.,  p.  444. 


MUTUAL   INDUCTION. 


419 


M\      R.( 

In  this  method  a  secohmmeter  may  be  used,  and  then 
the  galvanometer  terminals  will  be  reversed  at  each 
reversal  of  the  current.  Therefore,  a  dead  beat  galva- 
nometer may  be  substituted  for  the  ballistic  form,  and 


R2 


Fig.  192  b 

when  the  desired  condition  obtains,  it  will  indicate  that 
no  current  is  passing. 

109.  Coils  with  Iron  Cores.  —  When  measurements 
of  the  mutual-inductances  of  coils  with  iron  cores  are 
made  by  either  of  the  preceding  methods,  the  value 
observed  will  depend  upon  the  magnitude  of  the  cur- 
rent used  in  making  the  measurements.  Since  it  is  not 
practicable  to  use  currents  of  much  magnitude  in  the 
bridge  methods,  they  are  not  adapted  to  the  measure- 
ment of  the  mutual-inductances  of  coils  with  iron  cores 
which  are  designed  for  use  with  large  currents.  The 


420  ALTERNATING   CURRENTS. 

last  method  is  a  laborious  one,  and  therefore  is  not  well 
adapted  to  general  use  unless  modified  by  employing  a 
variable  standard,  as  described  below.  The  first,  sec- 
ond, and  third  methods  are  fairly  convenient,  and  may 
be  used  with  any  desired  current  in  the  primary  coil. 
They  are  also  fairly  reliable.  If  the  value  of  M  is  to 
be  determined  for  a  pair  of  iron-cored  coils  using  a 
certain  current,  it  is  only  necessary  to  adjust  the 
resistance  of  the  primary  circuit  so  that  the  required 
current  will  flow  when  the  key  is  closed.  The  stand- 
ards of  self-inductance  or  of  mutual-inductance  em- 
ployed in  the  comparative  methods  must  evidently  be 
constructed  without  iron  cores  so  that  the  coefficients 
are  independent  of  the  value  of  the  testing  current. 
A  variable  standard  of  mutual-inductance  may  be  made 
up  to  serve  a  purpose  similar  to  that  of  the  Ayrton 
and  Perry  self-inductance  standard  (Sect.  36,  3  a).  In 
fact,  the  Ayrton  and  Perry  self-inductance  standard 
may  be  used  as  a  variable  standard  mutual-inductance 
by  using  the  fixed  and  movable  coils  for  the  mutually 
interacting  pair,  in  which  case  the  mutual-inductance 
of  the  pair  may  be  varied  at  will  by  rotating  the 
movable  coil.  With  such  a  variable  standard  the  last 
method  enumerated  above  may  be  somewhat  simplified. 
In  this  case  the  adjustments  required  to  gain  a  balance 
may  be  made  by  changing  the  variable  standard.  If 
the  unknown  mutual-inductance  is  beyond  the  range 
of  the  standard,  the  balance  may  still  be  gained  by 

r> 

making  — 1  (page  99)  some  satisfactory  fixed  ratio  and 

R* 
then  balancing  by  adjusting  the  standard. 


MUTUAL   INDUCTION.  421 

110.   Mutual  Induction  of  Parallel  Distributing  Circuits. 

Where  two  or  more  electric  light  or  power  circuits 

carrying  alternating  currents  run  parallel  to  each  other, 
they  act  inductively  upon  each  other,  and  in  some  cases 
the  mutual  induction  may  cause  considerable  interference 
with  the  uniformity  of  the  pressure  on  the  lines.  The 
mutual  inductance  of  any  two  parallel  circuits  of  indefi- 
nitely great  length  may  be  easily  calculated,  provided 
the  distances  apart  of  the  different  wires  composing  the 
circuits  are  known.  The  number  of  lines  of  force  which 
pass  through  or  link  with  one  circuit,  due  to  one  ampere 
flowing  in  the  other,  is  numerically  equal  to  io9  times 
the  mutual-inductance  of  the  two  circuits,  and  this  num- 
ber of  lines  of  force  is  equal  to  the  algebraic  sum  of  the 
number  of  lines  of  force  embraced  by  the  first  circuit 
which  would  be  set  up  by  the  current  in  the  individual 
conductors  of  the  second  circuit  taken  separately.  The 
method  of  Section  47  is  therefore  directly  applicable  to 
the  calculation  of  the  mutual-inductance  of  two  long 
and  parallel,  narrow  circuits.  The  following  examples 
represent  the  commonest  arrangements  of  circuits  on 
pole  lines.  Suppose  that  a,  a'  and  b,  V  represent  the 
conductors  of  two  circuits,  and  that  the  order  of  the 
wires  is  a  —  af  —  b  —  b1 ,  the  distance  apart  centre  to 
centre  of  the  wires  of  circuit  A  is  x,  of  circuit  B  is  y, 
and  of  the  adjacent  wires  of  the  two  circuits  (a'  —  b)  is  z  ; 
then,  if  we  consider  the  currents  as  concentrated  at  the 
centres  of  the  wires,  which  makes  but  an  insignificant 
error  with  the  ordinary  dimensions  of  conductors  and 
circuits,  and  consider  the  space  between  two  planes  per- 
pendicular to  the  circuits  and  one  centimeter  apart,  the 


422  ALTERNATING   CURRENTS. 

number  of  lines  of  force  due  to  a  current  of  one  ampere 
in  ar,  which  pass  through  the  circuit  B  between  the 
planes,  is  (Sect.  47) 

y+*2  da 


=   I 

Jz 


and  th  j  number  of  lines  of  force  due  to  a  current  of  one 
ampere  in  a  which  pass  through  the  circuit  B  between 
the  planes,  is 


r 

=  -  \ 
J* 


*+y+*2da  .      x+ 

-  =  -  2loge 


The  total  number  of  lines  of  force  set  up  by  the  current 
of  one  ampere  in  circuit  A,  which  pass  through  the  cir- 
cuit B  between  the  planes,  is  Na  +  N^  the  number 
which  link  through  the  B  circuit  in  a  length  of  /  centi- 
meters is 


and  the  mutual-inductance  of  the  parallel  circuits  of 
length  /  is 


.        =      10  .      _  los> 

io9  io9V    '       z 


_ 

io9     ge 


=y,  this  becomes 

(^  +  ^)2          4.60  / 


M= 


and  \ix=y=z,  it  becomes 


MUTUAL   INDUCTION.  423 

where  /  is  the  length  of  the  parallel  circuits  in  centi- 
meters. 

Exchanging  the  order  of  the  wires  so  that  circuit  A  is 
between  the  conductors  of  circuit  B,  thus  b  —  a  —  a'  —  b', 
changes  the  formulas.  Here  the  algebraic  sum  of  the 
number  of  lines  of  force  set  up  by  the  circuit  A  which 
link  with  circuit  B,  is  equal  to  the  total  number  of  lines 
of  force  set  up  by  circuit  A  minus  the  number  passing 
backwards  through  b  —  a  and  a'  —  b'  .  Suppose  a  —  a1  is 
equal  to  x  and  b  —  a  and  a'  —  b1  are  each  equal  to  y, 
then  the  total  number  of  lines  of  force  set  up  by  one 
ampere  in  a  length  of  one  centimeter  of  circuit  A,  is 


(r  being  the  radius  of  the  conductor),  and  the  number 
of  lines  due  to  circuit  A,  which  pass  between  the  planes 
through  the  space  b  —  a,  is 


and 


If  the  circuits  are  not  in   the   same  plane,  as,  for 
instance,  they  are  arranged  thus, 


a  -a, 
b-b>. 


424  ALTERNATING   CURRENTS. 

and  the  distance  a  —  a'  is  x,  the  distance  b  —  b'  is  y,  the 
distance  a'  —  b'  is  s,  the  distance  a'  —  b  is  w,  a  —  b  is  v, 
and  a  —  b1  is  u ;  then  the  formulas  are 

u 

lo0"  — » 
36  V 

w      ,       z\  w 

and 


If  one  circuit  is  directly  beneath   the   other,  x=y, 
v  =  z,  and  w  =  u=  ^x*  +  ^2,  and  the  formula  becomes 


If-  2    Ice       +     -    -        , 

10g'  Ogl° 


These  results  plainly  show  that  the  mutual-inductance 
of  two  circuits  is  entirely  independent  of  the  actual  dis- 
tances apart  of  the  conductors  composing  the  circuits, 
but  depends  wholly  upon  the  relative  values  of  the 
distances.  The  mutual-inductance  of  two  circuits  is  a 
maximum  when  the  circuits  are  exactly  superposed,  in 
which  case  M=  ^/L'L"  —L,  and  decreases  as  the  dis- 
tance between  the  circuits  is  increased  in  comparison 
with  the  distance  apart  of  the  conductors  of  each  circuit  ; 
consequently,  mutual-inductance  between  circuits  on  the 
same  pole  line  may  be  reduced  by  decreasing  the  dis- 
tance apart  of  the  conductors  of  each  circuit  and  increas- 
ing the  distance  apart  of  the  circuits.  A  better  way  to 


MUTUAL   INDUCTION.  425 

avoid  mutual  induction  in  some  cases  is  to  transpose 
the  position  of  the  circuits  with  reference  to  each  other, 
as  is  done  in  long  distance  telephone  lines,  so  that  the 
inductive  effects  of  the  circuits  on  each  other  are  in 
opposition  in  different  parts  of  the  line,  and  neutralize 
each  other  for  the  line  as  a  whole. 

The  effect  of  mutual  induction  between  two  circuits 
is  to  set  up  an  electrical  pressure  in  one  when  the  cur- 
rent in  the  other  varies.  If  the  current  is  a  sinusoidal 
alternating  one,  this  pressure  is  (Sects.  107  and  108) 
2  7T/MC,  and  the  effect  of  an  alternating  current  in  one 
circuit  upon  another  circuit  is  easily  determined  if  M  is 
known.  When  the  two  circuits  are  fed  from  the  same 
single-phase  alternator,  the  induction  of  one  upon  the 
other  is  in  quadrature  with  the  current  in  the  first, 
and  the  relative  phase  of  the  pressure  induced  in  the 
second  depends  on  the  current  lag  in  the  first.  If  this 
is  zero,  the  induced  and  impressed  pressures  are  in 
quadrature,  while  they  are  in  opposition  if  the  lag  is 
90°.  The  result  is  a  displacement  of  the  pressure 
waves  and  a  drop  of  pressure  along  the  lines.  If  the 
circuits  are  fed  from  different  alternators,  the  frequency 
of  which  is  slightly  different,  the  inductive  pressure  and 
impressed  pressure  interfere  so  as  to  form  pulsations  or 
beats,  the  frequency  of  which  is  equal  to  the  difference 
of  the  two  alternator  frequencies,  and  the  amplitude  of 
which  is  the  sum  of  the  two  pressures.  This  may 
cause  a  perceptible  winking  of  incandescent  lamps  con- 
nected to  mutually  inductive  circuits  of  nearly  the  same 
frequency.* 

*  C.  F.  Scott,  Polyphase  Transmission,  Electrical  World,  Vol.  23,  p.  338  ; 
London  Electrician,  Vol.  32,  p.  642. 


426  ALTERNATING   CURRENTS. 


CHAPTER    X. 

OPERATION  OF  IDEAL  TRANSFORMER,  AND  EFFECT  OF 
IRON  AND  COPPER  LOSSES. 

111.   Ratio  of  Transformation   in   a   Transformer. — 

The  formulas  of  Section  104  show  that  the  electric  press- 
ure developed  in  the  secondary  coil  is  at  any  instant 

,,_ 


If  the  current  wave  is  a  sinusoid,  this  becomes 

ff  _  d(Mc* m  sin  a) 
~~dT 

If  the  conditions  require  that  M  be  treated  as  a  variable 
dependent  upon  the  varying  permeability  of  an  iron 
core,  this  equation  is  practically  unsolvable.  For  prac- 
tical purposes,  as  has  already  been  said,  it  is  sufficient 
to  assume  M  as  having  a  constant  average  value 
which  depends  upon  the  iron  of  the  core  and  the  mag- 
netic density  used  in  the  transformer.  The  equation 

..      Mc'm  cos  ada  , 

then  becomes  en  =  —      —. —    —     The  maximum  value 

at 

of  the  electric  pressure  is  therefore  e"m  =  2  7rfMc'm, 
where  /  is  the  frequency  of  the  current  wave.  The 
effective  value  of  the  secondary  electric  pressure  is  then 
evidently  E"  =  2  nrfMC',  where  C'  is  the  effective  pri- 


OPERATION   OF    IDEAL   TRANSFORMER.        427 

mary  current.       If  the  secondary  circuit   is  open,  the 
following  equations  may  be  written, 

rrt 

E 


while  E1.  =  2  TrfL'C'  =  V2  ^Nt  where  E'.  is  the  self- 

io8 

induced  primary  pressure,  E'  is  the  impressed  pressure, 
and  N  is  the  maximum  number  of  lines  of  force  in  the 
cycle.  If  the  resistance  of  the  primary  be  considered 
negligible,  the  former  equation  becomes 

C<=-#-, 

2  7T/Z' 

and  E'  =  27rfL'C  =  Efs. 

If  the  primary  and  secondary  coils  are  so  completely 
superposed  that  there  is  no  magnetic  leakage,  the  value 
of  M  becomes  J/=  VZ'Z,";  whence 

E'       27rL'C'          L'  .    &       VZ7 

'  l 


E"      2TT/MC'      VZ7Z77         E"      VZ77 

VZ7       n1 

But  —  '—  =  —-    since  the  reluctance  in  the  magnetic 
" 


circuit  of  the  two  coils  is  assumed  to  be  the  same  (Sect. 
16),  and  therefore 


In  other  words,  if  the  active  pressure  in  the  primary 
may  be  considered  negligible  when  compared  with  the 
impressed  pressure,  and  there  is  no  leakage  of  magnetic 
lines,  the  ratio  of  the  impressed  pressure  in  the  primary 
of  a  transformer  to  the  induced  pressure  in  the  sec- 


428         ALTERNATING  CURRENTS. 

ondary  is  equal  to  the  ratio  of  the  number  of  turns  of 
wire  in  the  two  coils.  This  ratio  of  pressures  is  called 
the  Ratio  of  Transformation.  The  ratio  of  transforma- 
tion of  well-designed  transformers  is  practically  equal 

to  —  when  the  secondary  circuit  is  open,  showing  that 

the  assumption  that  the  active  pressure  and  magnetic 
leakage  are  negligible  in  commercial  transformers,  when 
the  secondary  circuit  is  open,  is  entirely  allowable.  An 
example  will  show  this  in  a  striking  manner.  In  a 
certain  transformer  of  22.5  K.W.  capacity  the  resist- 
ance of  the  primary  is  practically  I  ohm  and  the  in- 
ductance is  9.1  henrys.  At  a  frequency  of  70  and  a 
pressure  of  2000  volts,  for  which  the  transformer  was 
designed,  the  value  of  47r2f2L'2  is  16,000,000.  In  an- 
other transformer  of  11.25  K.W.  capacity  designed  for 
2400  volts  primary  pressure,  the  value  of  the  primary 
resistance  is  6.45  ohms  and  the  value  of  47r2f2L'2  is 
10,000,000.  In  three  other  transformers  designed  for 
1000  volts  pressure  and  respectively  of  7.5,  4.5,  and  1.5 
K.W.  capacity,  the  primary  resistances  are  1.16,  2.15, 
and  8.90  ohms,  while,  at  a  frequency  of  125,  47r2f2L'2 
is  equal  respectively  to  100,000,000,  125,000,000,  and 
400,000,000;  and  in  a  transformer  of  .5  K.W.  capacity 
the  primary  resistance  is  25  ohms  and  47r2f2L12  is 
400,000,000.  In  each  of  these  cases,  which  represent 
common  practice  in  the  construction  of  transformers, 
the  value  of  R'2  is  entirely  negligible  when  compared 
with  47r2f2L'2.  If  R'2  were  not  negligible,  it  would 
evidently  increase  the  ratio  of  transformation  (that  is,  for 
a  given  impressed  primary  pressure  the  secondary  press- 


OPERATION  OF  IDEAL  TRANSFORMER.    429 

lire   would   be   decreased)   on    account    of   the   loss   of 
pressure  due  to  the  current  flowing  through  R' . 

112.  Magnetic  Leakage.  — The  primary  and  secondary 
coils  in  each  of  these  cases  are  so  sandwiched  together 
that  magnetic  leakage  is  certainly  negligible  when  there 
is  no  current  in  the  secondary  coil.  A  case  when  leak- 
age is  always  present  is  shown  in  Fig.  193.  From  the 
figure  it  is  evident  that  if  there  is  no  current  flowing  in 
the  secondary,  the  counter  pressure  in  the  primary  will 


a,  a,  a,  a.   LEAKAGE  LINES. 

b,  b.    USEFUL  LINES. 

Fig.  193 

be  greater  per  turn  of  wire  than  the  pressure  induced 
in  the  secondary  per  turn ;  hence,  as  the  self-induced  or 
counter  pressure  in  the  primary  is  practically  equal  and 
opposite  to  the  impressed  pressure,  the  ratio  of  trans- 
formation will  be  increased.  If  a  current  flows  in  the 
secondary,  the  self-induction  due  to  magnetic  leakage 
in  the  secondary  (lines  of  force  linking  with  the  sec- 
ondary coil  but  not  linking  with  the  primary  coil)  will 
still  further  reduce  the  active  secondary  pressure,  and 
the  ratio  of  transformation  will  be  further  increased. 


430  ALTERNATING   CURRENTS. 

The  effect  of  magnetic  leakage  in  increasing  the  ratio 
of  transformation  (decreasing  the  proportional  pressure 
induced  in  the  secondary  by  decreasing  the  magnetic 
induction  passing  through  it)  is  shown  by  an  experi- 
ment reported  by  Professor  Ryan.*  In  the  experi- 
ment recorded  by  him,  the  primary  and  secondary  coils 
were  wound  on  opposite  sides  of  a  laminated  iron  ring 
(Fig.  194).  The  number  of  turns  on  the  primary  and 

secondary  were  respectively  500  and   155,  or  -^  =  3.2. 


Pig.  194 

When  a  pressure  of  75.6  volts  was  impressed  upon  the 
primary  with  the  secondary  open,  a  pressure  of  only 
16.4  volts  was  induced  in  the  secondary,  or  —  =  4.6. 

The  whole  difference  in  the  two  ratios  was  due  to 
magnetic  leakage,  and  the  magnitude  of  the  difference 
shows  that  M  was  much  less  than  VZ7Z77.  In  fact, 
—  _.  1.447?  an(j  assummpr  that  the  reluctance  of  the 

VZ'Z77 
magnetic  circuits  of  the  two  coils  were  equal,  M=  — 

*  Some  Experiments  upon  Alternating  Current  Apparatus,  Trans.  Anier. 
Inst.  E.  E.,  Vol.  7,  p.  324. 


OPERATION  OF  IDEAL  TRANSFORMER.    431 

(by  Sect.  104).  The  magnetic  leakage  was  therefore 
30  per  cent ;  that  is,  the  number  of  lines  of  force  that 
passed  through  the  primary  but  not  through  the  sec- 
ondary coil,  was  30  per  cent  of  the  total  magnetic 
induction  set  up  in  the  magnetic  circuit.  The  effect 
of  magnetic  leakage  on  a  transformer  is  analogous  to 
the  effect  produced  on  one  without  leakage,  of  insert- 
ing coils  having  self-inductance,  or  Impedance  Coils,  in 
the  primary  and  secondary  circuits  outside  of  the  trans- 
former (Fig.  195).  These  coils  would  have  such  self- 

TRANSFORMER 


Fig.  195 

inductances  as  to  increase  the  self-inductances  of  the 
primary  and  secondary  circuits  in  the  ratio  of  a  :  100, 
where  a  is  the  magnetic  leakage  in  per  cent.  Since 
leakage  causes  a  proportional  increase  in  the  apparent 
self-inductance  of  the  primary  and  secondary  circuit,  it 
causes  an  equivalent  lag  of  the  currents  in  the  two 
circuits. 

113.  Exciting  Current.  —  In  the  case  of  an  ideal  trans- 
former without  losses,  the  lag  of  the  primary  current, 
when  the  secondary  circuit  is  open,  is  90°  with  respect 
to  the  impressed  pressure  ;  for,  tan  $  2 


(Sect. 


432  ALTERNATING   CURRENTS. 

28),  and  R'  is  assumed  to  be  zero.  Since  the  induced 
secondary  pressure  lags  behind  the  magnetism,  which 
is  in  phase  with  the  primary  current  when  the  sec- 
ondary circuit  is  open,  by  an  angle  of  90°,  the  phases 
of  the  primary  impressed  pressure  and  the  secondary 
induced  pressure  are  exactly  180°  apart,  or  they  are 
in  exact  opposition.  The  current  in  the  primary  cir- 
cuit of  an  ideal  transformer,  when  the  secondary  is 
open,  is  all  wattless,  and  of  a  magnitude  which  depends 
only  upon  the  self-inductance,  U ',  of  the  primary  coil. 

The  losses  due  to  hysteresis  and  foucault  currents 
in  the  iron  core  and  resistance  in  the  primary  coil  are 
by  no  means  negligible  in  commercial  transformers, 
but  are  of  such  a  magnitude  as  to  decrease  the  lag 
of  the  primary  current  until  the  power  factor  of  the 
primary  circuit  is  ordinarily  between  50  per  cent  and 
80  per  cent,  when  there  is  no  current  in  the  secondary 
circuit ;  but  the  magnetism  in  the  core  remains  in  phase 
with  the  wattless  component  of  the  primary  current 
and  is  90°  behind  the  phase  of  the  primary  pressure, 
so  that  the  primary  and  secondary  pressures  are  still 
in  opposition.*  The  current  which  flows  in  the  primary 
circuit  when  the  secondary  circuit  is  open,  may  there- 
fore be  considered  as  composed  of  two  components, 
one  of  which  supplies  the  energy  required  to  make  up 
the  transformer  losses,  and  the  other  of  which  serves 
simply  for  magnetizing  power,  and  is  therefore  wattless. 
This  primary  current  is  often  called  the  Leakage  Cur- 

*  Fleming,  Experimental  Researches  on  Alternate  Current  Transformers, 
Jour.  Inst.  E.  £.,  1892.  Ford,  Tests  of  Modern  Transformers,  Bull. 
Univ.  of  Wisconsin,  Vol.  I,  No.  n. 


OPERATION   OF    IDEAL   TRANSFORMER.         433 

rent  or  Open-Circuit  Current,  but  a  more  satisfactory 
term  is  Exciting  Current.  The  term  Magnetizing  Cur- 
rent is  also  applied  to  the  exciting  current,  but  we 
will  reserve  the  term  for  its  wattless  component  which 
is  truly  a  magnetizing  current. 

114.  Core  Magnetization.  —  The  maximum  magnetic 
induction  during  a  period,  in  the  iron  core  of  an  ideal 
transformer,  is  dependent  upon  the  maximum  value  of 
the  current  in  the  primary  circuit,  the  effect  of  iron 
losses  being  omitted  by  assumption,  and  is  equal  to 

»'Cy  f-n'CS  n'C' 


where  C\  is  the  magnetizing  current  and  P  is  the  reluct- 
ance of  the  magnetic  circuit  at  the  time  that  the  current 
has  its  maximum  value. 

Closing  the  secondary  circuit  so  that  a  current  may 
flow  in  it  under  the  impulse  of  the  induced  secondary 
pressure,  materially  changes  the  conditions  heretofore 
explained.  We  will  first  assume  that  the  secondary  cir- 
cuit is  without  self-inductance,  and  continue  to  neglect 
hysteresis  and  foucault  current  losses  in  the  iron  core 
and  resistance  losses  in  the  windings,  in  which  case 
the  secondary  current  C"  will  be  in  unison  with  the 
induced  or  secondary  pressure,  E"  .  This  current  has 
its  own  magnetizing  effect  on  the  magnetic  circuit.  If 
c'  and  cn  be  the  primary  and  secondary  currents  at 
any  instant,  the  total  magnetizing  force  in  the  circuit  is 

Ll*  =  jy.p,  where  JVt  is  the  instantaneous 

value  of  the  magnetic  induction,  and  P  is  the  assumed 
constant  value  of  the  reluctance  of  the  magnetic  circuit. 

2F 


434  ALTERNATING   CURRENTS. 

From  this  is  found 

n"c 


,      10        m   . 
and  whence        c  =  -  -^smo,  --  -  -  cos  a; 
4  TTH 


but 


whence  c'  =  -\/2[  C-J  sin  a  —  ~C"  cos  a) 

V  W  ) 

or  n'c'  =  -V2  (n1  C^  sin  a  —  n"  C"  cos  a), 

and  n^C*  =  -£$*€£  sin2  a 

—  2  n'n"CiC"  sin  a  cos  a 


where  C'  is  the  effective  value  of  the  primary  current 
when  the  secondary  current  is  equal  to  C11,  and  CJ,  as  be- 
fore, is  the  wattless  primary  current  when  the  secondary 
is  open.  Performing  the  integration  gives 


Remembering  that  CJ  and  Cn  have  90°  difference  of 
phase,  the  three  terms  of  this  formula  may  be  repre- 
sented by  the  three  sides  of  a  right-angled  triangle 
(Fig.  196).  The  current  C  ',  which  flows  in  the  pri- 
mary when  the  secondary  is  loaded,  is  in  advance  of 
the  current  £/  by  an  angle  i/r,  the  tangent  of  which  is 

shown  by  the  figure  to  be   tan  i/r  =  n        -     We  have 

H  C-i 

already  seen  that  C±    is  inversely  dependent  upon  the 


EFFECT  OF  COPPER  AND  IRON  LOSSES.   435 


self-inductance  of  the  primary  circuit,  and  therefore — 
when  the  value  of  n1  is  fixed — directly  upon  the  re- 
luctance of  the  magnetic  circuit,  which  in  commercial 
transformers  is  made  very  small.  Tan  ty  is  therefore 
quite  large  when  Cn  has  any  considerable  magnitude 


O 


"T'A 


B 


Fig.  196 


and  t/r  approaches  90°  as  C"  increases,  so  that  the  pri- 
mary and  secondary  currents  are  practically  in  opposite 
phases  in  a  well  loaded  transformer.  As  a  transformer 
is  loaded  up,  its  power  factor  is  therefore  rapidly  in- 
creased.* The  effect  of  the  secondary  current  on  the 

*  Compare  Fleming,  Experimental  Researches  on  Alternate  Current 
Transformers,  Jour.  Inst.  E.  £.,  Vol.  21,  p.  594;  Ford,  Tests  of  Modern 
Transformers,  Bull.  Univ.  of  Wisconsin,  Vol.'i,  No.  II. 


436  ALTERNATING   CURRENTS. 

primary  circuit  is  to  apparently  decrease  its  self-induc- 
tance and  therefore  to  decrease  its  impedance  and  the 
lag  of  the  primary  current. 

115,  Effect  of  Copper  and  Iron  Losses  on  Regulation. 
—  Consideration  of  the  effect  of  C2R,  hysteresis,  and 
foucault  current  losses  has  thus  far  been  neglected,  but 
it  has  been  shown  that  the  effects  of  these  losses  are 
by  no  means  negligible.  It  is  shown  in  Section  1 1 1 
that  the  effect  of  the  primary  resistance,  Rf,  is  to  cause 
a  fall  in  the  secondary  pressure  and  therefore  to  in- 
crease the  ratio  of  transformation.  The  resistance  of 
the  secondary  winding,  R" ,  evidently  acts  to  cause  a 
decrease  in  the  pressure  at  the  terminals  of  the  second- 
ary and  therefore  to  increase  the  apparent  ratio  of  trans- 
formation. The  magnitude  of  the  apparent  change  in 
the  ratio  of  transformation  is  dependent  upon  the  sum 
of  the  products  of  the  resistances  with  the  currents  in 
the  respective  circuits;  that  is,  to  the  pressure  required 
to  pass  the  current  through  the  resistances  of  the  cir- 
cuits. The  loss  of  pressure  at  the  secondary  terminals 
in  volts,  due  to  this  cause,  when  current  C"  flows  in  the 
secondary  circuit  is 

V=C"R"  +^C'R', 

«  w< 
and  since  approximately,  with  core  losses  neglected, 

C"=—rf 

n"C  ' 

this  is  V=  C"\R"  +  ( 

L  \ 

The  percentage  increase  of  the  ratio  of  transformation 


EFFECT    OF   COPPER   AND    IRON    LOSSES.       437 

is  — -,  where  E"  is  the  total  secondary  pressure,  the  ter- 
minal pressure  becoming  E" —  V.  This  shows  that  the 
terminal  pressure  falls  off  proportionally  as  the  load  on 
the  secondary  of  the  transformer  is  increased,  if  the 
impressed  primary  pressure  remains  constant.  The 
formula  also  shows  that  an  ideal  transformer  (i.e.  one 
without  resistance,  core  losses,  or  magnetic  leakage)  is 
inherently  self-regulating,  and  will  therefore  give  a  con- 
stant pressure  at  the  secondary  terminals  at  all  loads  if 
fed  with  a  constant  primary  pressure. 

The  effect  of  core  losses  (hysteresis  and  foucault 
current  losses)  is  to  increase  the  primary  current  to  a 
certain  extent  and  therefore  to  slightly  affect  the 
regulation. 

116.  Perfect  Regulation  of  an  Ideal  Transformer. — The 
statements  of  the  preceding  section  may  also  be  proved 
as  follows  :  Considering  the  phases  of  the  primary  cur- 
rent and  magnetization  to  be  practically  90°  apart,  then 
the  counter  electric  pressure  of  self-induction  is  in 
opposition  to  the  impressed  pressure,  and  the  active 
pressure  in  the  primary  circuit  at  the  instant  when  the 
impressed  pressure  is  a  maximum  is 

F'    -  -  F'     —  rf  /?' 

J2>   m          •£*   sm  —  C  -K-   > 

E',  being  the  counter  electric  pressure  of  self-induction. 
At  this  instant  the  value  of  the  primary  current,  cf,  is 


nn  /" 
since  (Sect.  114)  sin  ^  = 

n'C' 


u'C 
ft  /"tr 


438  ALTERNATING   CURRENTS. 

The  counter  electric   pressure  of  self-induction  is  evi- 
dently equal  to 


wnere  R  is  the  external  resistance  in  the  secondary 
circuit.  Substituting  these  values  for  c'  and  E1  ',  and 
dividing  by  V2  gives 

E'  =  ^~  OR1  +  4  C"  (R"  +  ^)> 
n'O  n" 

whence,  by  transposition, 


C"R=—E'  -C'R1  -  C"R". 
n'  \n'  J 

From  these  equations  it   is  seen  that  the   pressure  at 

E'n" 
the  secondary  terminals,  C"  R,  becomes  —  -—  provided 

n 
and  Rn   can  be  taken  as  very  small  in  com- 


fn"\z 

{  —  7  )  R1 

\n!  J 


parison  with  R,  and  therefore  under  these  circum- 
stances the  secondary  pressure  is  constant  provided  the 
impressed  pressure  be  kept  constant.  An  ideal  trans- 
former is  therefore  an  inherently  self  -regulating  instru- 
ment for  transforming  electric  currents  at  one  constant 
pressure  into  equivalent  currents  at  another  constant 
pressure,  and  the  faulty  regulation  found  in  commercial 
transformers  is  wholly  dzie  to  electrical  losses  and  mag- 
netic leakage.  By  transforming  the  last  formula  into 
the  equivalent  form  C"  =C'  -^y,  it  is  seen  that  an  ideal 

transformer  which  is  fed  with  a  constant  current  is  an 
inherently  self-regulating  instrument  for  the  transforma- 
tion of  that  current  into  an  equivalent  constant  current 


EFFECT  OF  COPPER  AND  IRON  LOSSES.   439 

at  another  pressure.  In  this  case  the  primary  impressed 
pressure  will  vary  with  the  resistance  of  the  secondary 
circuit. 

117.  Effect  on  Regulation  of  Self-inductance  or  Capacity 
in  Secondary  Circuit.  —  In  the  service  to  which  trans- 


Fig.  197 

formers  have  heretofore  been  generally  applied,  the 
operation  of  incandescent  lamps,  the  external  secondary 
circuit  is  practically  non-inductive,  but  when  motors  or 
arc  lamps  are  operated  on  the  secondary  circuits  of 
transformers,  they  may  add  a  considerable  inductance 


440  ALTERNATING   CURRENTS. 

to  the  circuits.  In  this  case  the  secondary  current  is 
caused  to  lag  behind  the  secondary  pressure.  The  rela- 
tion which  must  exist  between  the  secondary  and  primary 
ampere-turns  and  the  resultant  magnetizing  ampere-turns 
(Fig.  197)  shows  that  such  a  lag  of  the  secondary  cur- 
rent must  cause  an  increase  in  the  lag  of  the  primary 
current.  The  effect  is  exactly  as  though  additional  self- 
inductance  were  placed  in  the  circuit  of  the  primary 
coil,  and  an  inductive  external  secondary  circuit  there- 
fore causes  defective  regulation  on  the  part  of  the 
transformer.  The  result  is  an  increase  in  the  ratio  of 
transformation  which  depends  upon  the  resistance  and 

reactance  of  the  secondary  circuit,  since  tan  <£  =  — ^ — • 

The  effect  of  a  capacity  in  the  secondary  circuit  is 
exactly  opposite  to  that  of  an  inductance,  since  it 
causes  the  current  to  lead  the  pressure.  Consequently, 
a  secondary  circuit  having  capacity  tends  to  aid  regu- 
lation, and  may,  if  the  capacity  is  sufficient,  even  cause 
a  decrease  in  the  ratio  of  transformation.  That  is,  the 
pressure  in  a  secondary  circuit  may  be  increased  by 
the  mere  insertion  of  a  condenser. 

118.  Graphical  Method  for  Determining  Current  and 
Pressure  Relations.  —  The  effects  discussed  may  all  be 
shown  very  plainly  by  a  graphical  construction  based 
upon  the  triangle  of  electrical  forces  (Sect.  15).  In 
Fig.  198,  OC"  on  the  vertical  axis  represents  the  value, 
on  a  convenient  scale,  of  the  product  n"C".  If  the 
secondary  circuit  may  be  considered  non-inductive,  as 
when  the  transformer  is  feeding  incandescent  lamps, 
the  secondary  pressure  wave  is  in  unison  with  the  cur- 


EFFECT   OF   COPPER   AND    IRON    LOSSES.       441 


rent,  and  OE"   may  be  taken  to  represent  the  value 
and  position  of  the  pressure.     The  current  component 


Pig-.  198 

which  is  effective  in  producing  magnetization  must  be 
90°  in  advance  of  this.  Accepting  the  conventional 
positive  direction  for  harmonic  rotation  as  left-handed 


442  ALTERNATING  CURRENTS. 

or  counter-clockwise,  the  magnetizing  ampere-turns  «'£/ 
must  be  laid  off  on  the  horizontal  line  to  the  right  of 
the  vertical  and  may  be  represented  by  OC^.  The  am- 
pere-turns of  the  primary,  when  C"  flows  in  the  second- 
ary, are  found  by  completing  the  parallelogram  on  OC" 
of  which  OCi  is  the  diagonal.  This  gives  the  line  OC' 
to  represent  n' C' .  In  order  that  the  diagram  may  be 
readily  intelligible  the  value  of  n'C^  is  taken  as  about 
|  of  n'fCff,  while  in  commercial  transformers  it  is  gen- 
erally less  than  ^  of  n"C"  and  is  sometimes  as  small 
as  -5*0  or  g*Q  of  n" C" .  The  angle  YOC'  in  the  diagram 
is  therefore  much  exaggerated  in  comparison  with  its 
value  in  commercial  transformers. 

It  now  remains  to  find  the  value  and  position  OJL  tne 
impressed  primary  pressure.  This  is  the  resultant  of 
the  counter  pressure  of  self-induction  in  the  primary 
circuit,  EJ,  and  the  active  pressure  —  the  pressure  which 
is  effective  in  making  up  the  losses  in  the  magnetic 
circuit  caused  by  hysteresis  and  foucault  currents  and 
in  the  conductors  of  the  primary  coil  caused  by  its  re- 
sistance. The  second  component  of  the  pressure  is  in 
unison  with  the  primary  current  and  may  be  laid  off  on 
the  line  OC,  its  length  being  OEW.  The  self-inductive 
primary  pressure  is  in  unison  with  and  in  the  same 
direction  as  the  induced  secondary  pressure,  and  is 

equal  to  Eh r~  if  M=^L'L".     It  is  represented  in  the 

figure  by  OEJ.  Completing  the  parallelogram  gives 
the  line  OE1,  which  represents  the  direction  and  magni- 
tude of  the  impressed  electric  pressure.  In  the  figure 
the  angle  E' OC  =  </>',  and  the  angle  COQ  =  f .  The 


EFFECT   OF   COPPER   AND  'IRON   LOSSES.       443 

relative  phases  of  the  pressures  and  currents  are  shown 
by  the  relative  angular  positions  of  the  lines  radiating 
from  O.  The  value  of  the  primary  current  is  taken 
directly  from  the  length  of  the  line  OC'.  When  the 


/£„  C' 


£-f-     Fig.  199 

secondary  circuit  is  open,  the  construction  is  similar  to 
the  preceding,  but  the  value  of  the  primary  CR  loss 
is  less,  making  the  length  of  OE'  slightly  smaller  (Fig. 
199).  The  exciting  current  is  taken,  as  before,  directly 


444 


ALTERNATING  CURRENTS. 


from  the  length  of  the  line  OC' ,  and  the  figures  show 
that  the  ratio  of  transformation  is  increased  by  loading 
the  transformer  on  account  of  the  drop  of  pressure  in 


the  windings.  The  figures  plainly  show  that  the  devi- 
ation of  the  secondary  pressure  from  the  form  of  the 
primary  pressure  is  proportional  to  the  value  of  C' R* . 


EFFECT  OF  COPPER  AND  IRON  LOSSES.   445 


The  hysteresis  and  foucault  current  losses  may  be 
assumed  to  be  independent  of  the  secondary  current 
(Sect.  127). 


The  effect  of  inductance  in  the  external  secondary 
circuit  is  shown  in  Fig.  200.  As  before,  OCf  repre- 
sents the  secondary  ampere-turns  nflC".  If  it  is  sup- 


446  ALTERNATING   CURRENTS. 

posed  that  the  inductance  in  the  secondary  circuit  be 
sufficient  to  cause  a  lag  of  <f>",  then  the  induced  second- 
ary pressure  is  in  advance  of  the  current  by  an  angle 
<f>",  and  is  represented  in  magnitude  and  direction  by 
OEff.  The  position  of  the  magnetizing  ampere-turns 
is  90°  in  advance  of  OE" ',  and  is  represented  by  OC-^. 
Completing  the  parallelogram  on  OC'f  and  OC^,  gives 
OCr.  The  primary  impressed  pressure  is  then  found 
as  before,  and  a  comparison  of  Figs.  198  and  200  shows 
that  the  self-inductance  in  the  secondary  circuit  in- 
creases <j>'. 

A  similar  construction,  showing  the  effect  of  capacity, 
is  given  in  Fig.  201.  This  differs  from  the  preceding 
only  on  account  of  the  secondary  current  leading  the 
pressure. 

119.  Transformation  from  Constant  Pressure  to  Con- 
stant Current.  —  The  effect  of  magnetic  leakage  can 
also  be  satisfactorily  shown  in  the  same  manner  (Fig. 
202).  Remembering  that  the  effect  of  leakage  is  the 
same  as  that  of  self-inductance  coils  placed  in  the  pri- 
mary .and  secondary  circuits,  the  construction  is  exactly 
the  same  as  in  the  case  of  a  transformer  working  on  an 
inductive  secondary  circuit,  with  an  additional  correction 
applied  to  the  angle  of  lag  between  the  primary  press- 
ure and  current  to  account  for  the  direct  effect  of  the 
leakage  on  the  primary  circuit.  The  construction  shows 
that,  as  the  leakage  is  increased  so  that  the  secondary 
angle  of  lag  <j>"  approaches  90°,  the  deficiency  in  the 
inherent  tendency  to  regulate  for  constant  pressure  be- 
comes so  great  that  the  secondary  terminal  pressure 
actually  tends  to  vary  inversely  with  the  current.  Such 


EFFECT   OF   COPPER   AND    IRON    LOSSES.       447 


44* 


ALTERNATING   CURRENTS. 


a  transformer  would  therefore  tend  to  transform  a  vari- 
able current  at  constant  pressure  into  a  constant  current 
at  a  variable  pressure,  which  would  enable  it  to  be  used 
for  series  arc  lighting  from  a  constant-pressure  circuit. 
When  the  lag  angle  becomes  90°,  the  transformer  can 
of  course  do  no  work,  consequently  it  is  impossible 
to  get  very  exact  regulation  in  thus  transforming  from 
constant  pressure  to  constant  current,  but  it  is  possible 


Fig.  2O3 

to  arrange  the  transformer  so  that  the  percentage  can 
be  varied  when  necessary  by  partially  closing  a  shunt 
magnetic  circuit  by  a  slab  of  iron  strips,  as  was  first  pro- 
posed by  Elihu  Thomson  (Fig.  203).  Figure  204  shows 
the  results  of  a  test  of  a  Wood  transformer,  in  which 
the  constant-current  regulation  is  wholly  due  to  mag- 
netic leakage.  In  the  upper  half  of  the  figure,  one 
curve  shows  the  efficiency  as  a  function  of  the  current 


EFFECT  OF  COPPER  AND  IRON  LOSSES.   449 


in  the  secondary  circuit,  and  the  other  curve  shows  the 
external  characteristic,  or  the  secondary  terminal  press- 


coo 


500 


400 


H300 


L'OO 


100 


AMPERES 


468 
Fig.  204 


.10 


12 


10 


ure  as  a  function  of  the  secondary  current.     The  lower 
half  of  the  figure  has  curves  which  show  the  watts  in 

2G 


450  ALTERNATING   CURRENTS. 

the  primary  and  secondary  circuits  as  a  function  of  the 
secondary  current.  The  crosses  on  the  curves  show 
the  points  corresponding  to  normal  load,  which  is  that 
required  to  operate  one  arc  lamp.  The  primary  press- 
ure of  this  transformer  was  1000  volts. 

When  magnetic  leakage  makes  itself  evident  in  a 
transformer  with  the  secondary  circuit  open,  the  pre- 
ceding equations  relating  to  constant-pressure  regula- 
tion are  vitiated,  since  the  ratio  of  transformation  is 

i 

no  longer  equal  to  — .  In  well-built  transformers  de- 
signed for  constant  pressure,  magnetic  leakage  is  not 
likely  to  be  of  much  magnitude,  and  in  fact  it  can  only 
be  brought  to  a  large  value  by  making  the  space  occu- 
pied by  the  primary  and  secondary  coils  very  large 
compared  with  the  cross-section  of  the  iron  core,  by 
using  iron  of  a  low  permeability,  or  by  specially  arrang- 
ing leakage  paths. 

120.  The  Effects  of  Variable  Reluctance,  Hysteresis, 
and  Foucault  Currents  on  the  Form  of  the  Primary  Cur- 
rent Wave.  —  In  the  preceding  discussions  it  has  been 
assumed  that  the  reluctance  of  the  magnetic  circuits  of 
transformers  can  be  taken  at  an  average  constant  value 
which  is  practically  equal  to  that  when  the  current  is 
at  its  maximum  point.  The  low  induction  which  is 
used  in  commercial  transformers  as  ordinarily  con- 
structed, makes  this  assumption  entirely  allowable, 
though  it  is  by  no  means  exact.  If  the  induction  be 
pushed  above  the  bend  in  the  curve  of  magnetization, 
however,  the  influence  of  the  lowered  permeability  of 
the  iron  becomes  marked.  The  curve  CM  in  Fig.  205 


EFFECT  OF  COPPER  AND  IRON  LOSSES.   451 

may  be  taken  to  represent  the  curve  of  magnetization 
of  a  transformer  core,  plotted  with  ampere-turns  as 
abscissas  and  volts  induced  in  the  primary  windings 


Fig-.  2O5 

as  ordinates,  supposing  the  effect  of  hysteresis  to  be 
negligible.  Then  when  the  magnetizing  turns  equal 
n'C},  the  induced  pressures  in  the  primary  and  second- 
ary circuits  are  reduced  from  EJ  and  E",  which  would 
be  reached  with  a  constant  reluctance,  to  Els'  and  E^'. 


452  ALTERNATING   CURRENTS. 

The  construction  shows  that  this  decreases  the  ansrle  of 

o 

lag  between  the  primary  current  and  pressure,  and  makes 
necessary  more  turns  of  wire  on  the  primary  and  sec- 
ondary coils  in  order  that  a  given  output  may  be 
obtained.  Since,  in  this  case,  the  permeability  varies 
through  each  period  with  the  magnetizing  ampere- 
turns,  there  is  a  periodic  variation  of  <£/,  and  the 
primary  current  wave  is  distorted  from  the  form  of 
the  primary  impressed  pressure  wave. 

The  effect  of  hysteresis  in  the  core  of  a  transformer 
is  to  distort  the  form  of  the  primary  current  wave  to 
a  still  more  marked  degree  than  would  magnetic  satu- 
ration alone,  and  the  higher  the  maximum  magnetic 
density  is  carried,  the  greater  the  distortion  becomes. 
The  ordinates  of  the  primary  current  wave  are  at  each 
instant  proportional  to  the  difference  of  the  correspond- 
ing ordinates  of  the  primary  pressure  wave*  and  the 
wave  of  counter  electric  pressure.  The  latter  is  of 
course  exactly  similar  to  the  form  of  the  secondary 
pressure  wave.  With  the  primary  pressure  wave  sinusoi- 
dal and  an  approximately  uniform  magnetic  reluctance, 
the  primary  current  wave  would  be  sinusoidal.  With  a 
variable  reluctance,  but  no  hysteresis,  the  current  wave 
becomes  peaked,  but  remains  symmetrical ;  but  when 
hysteresis  is  taken  into  account,  the  symmetrical  form 
is  lost.  This  is  conveniently  illustrated  by  Figs.  206 
and  207.  In  Fig.  206  the  heavy  line  represents  the 
hysteresis  cycle  for  a  piece  of  wrought  iron,  and  the  line 
b'Ob  may  be  taken  to  represent  the  cyclic  curve  of  mag- 
netization which  would  be  given  by  the  iron  if  hyste- 
resis were  absent.  In  Fig.  207  the  curve  M  represents 


EFFECT   OF   COPPER   AND   IRON   LOSSES.       453 


the  curve  of  magnetic  induction  in  a  transformer  with  its 
ordinates  plotted  on  the  same  scale  as  Fig.  206.  When 
the  transformer  is  worked  with  its  secondary  circuit  open, 
the  drop  of  pressure  due  to  the  exciting  current  flowing 
through  the  primary  winding  is  negligible,  so  that  the 
primary  impressed  pressure  at  each  instant  is  propor- 
tional to  the  tangent  of 
the  curve  of  magnetism 
and  is  90°  in  advance  of 
the  magnetization.  In 
this  case,  the  curves  of 
pressure  and  magnetism 
are  assumed  to  be  sinu- 
soidal. The  tangent 
relations  between  the 
curves  of  magnetization 
and  pressure  (Sect.  77) 
must  exist  as  long  as 
C'R'  is  negligible.  As 
an  illustration  of  the 
fact  that  the  self-in- 
duced pressure  is  of 
the  same  form,  and  is  Figr-  2O0 

equal  and  opposite  to  the  impressed  pressure,  and  that 
the  exciting  current  varies  in  such  a  way  as  to  furnish 
this  induced  pressure  in  a  transformer  with  the  second- 
ary open  (supposing  the  C2R  losses  negligible),  we 
may  consider  an  inductance  coil  of  negligible  resist- 
ance with  a  sinusoidal  pressure  (E)  of,  say,  100  volts 
impressed  upon  it.  Suppose  the  frequency  is  127^,  the 
self-inductance  (Zj)  is  .01  of  a  henry,  and  the  resistance 


454         ALTERNATING  CURRENTS. 

is  negligible  ;   then   2  7rfLl  =  8  will  be   the  impedance 


(see  Chap.  IV.).  The  current  (£\)  flowing  will  be 
-  —=12.5  amperes  with  a  lag  angle  of  90°.  The 

27T/L1 

inductive  pressure  (EJ  will  be  2  7rfLlCl  =  100  volts, 
which  is  equal  and  opposite  to  the  impressed  pressure. 
Suppose  the  self-inductance  Ll  change  to  Z2  =  .005  ; 
then  the  current  will  be 


=  25, 


27T/Z, 

and  the  inductive  pressure  (E2)  will  be 
2  7rfL2C2  =  100, 

as  before.  It  is  seen  that  whatever  value  the  self- 
inductance  may  have,  the  exciting  current  will  take 
such  a  magnitude  that  the  counter  pressure  of  self- 
induction  will  be  equal  and  opposite  to  the  impressed 
pressure.  As  the  reluctance  of  the  magnetic  circuit 
is  proportional  to  the  self-inductance,  when  the  reluc- 
tance changes,  the  exciting  current  flowing  will  change 
so  that,  as  before,  the  counter  pressure  will  equal  the 
impressed  pressure.  Now,  neglecting  hysteresis,  when 
the  magnetism  is  carried  through  the  cycle  ObOb'O 
(Fig.  206),  the  current  corresponding  to  each  ordinate 
of  the  curve  M  in  Fig.  207  must  be  proportional  to 
the  current  required  to  produce  the  corresponding  mag- 
netization in  Fig.  206.  In  this  way  curve  Ce  is  plotted 
to  represent  the  wave  of  exciting  current  in  the  trans- 
former if  there  were  no  hysteresis  in  the  iron.  This 
curve  is  symmetrical  and  of  the  same  phase  as  curve  M, 
and  it  represents  the  true  wattless  magnetizing  current. 


EFFECT   OF   COPPER   AND    IRON    LOSSES.       455 


It  is  easy  to  plot  the  curve  which  shows  the  form 
of  exciting  current  of  the  transformer  with  the  effect 
of  hysteresis  included,  since  it  is  only  necessary  to  plot 
as  current  ordinates  in  Fig.  207  the  currents  which 
correspond  to  the  various  values  of  the  magnetization 
in  the  hysteresis  cycle.  This  curve  is  curve  C  in  Fig. 
207.  It  may  be  considered  as  made  up  of  a  wattless 
magnetizing  component,  Ce,  and  an  active  component 


/ 


C.\ 


Fig.  2O7 

Ch  caused  by  hysteresis.  The  ordinates  of  the  active 
component  are  proportional  to  the  differences  of  cor- 
responding abscissas  of  the  curve  of  magnetization  and 
the  hysteresis  cycle  (Fig.  206),  and  the  curve  is  nearly 
sinusoidal.  The  co-ordinates  of  the  points  b  and  b' 
in  Fig.  206  are  not  altered  by  hysteresis  and  the  effect 
of  hysteresis  therefore  does  not  alter  the  position  or 
magnitude  of  the  maximum  value  of  the  exciting  cur- 


456 


ALTERNATING   CURRENTS. 


rent,  though  it  distorts  and  enlarges  the  loop  of  the 
curve,  and  advances  its  zero  points  slightly  so  that 
the  equivalent  lag  angle  becomes  slightly  less  than  90°. 
The  effect  which  foucault  currents  in  the  core  have 
upon  the  exciting  current  may  be  shown  in  a  similar 
manner.  The  instantaneous  value  of  the  portion  of  the 
exciting  current  which  is  required  to  make  up  the  losses 


\ 


i        o 

AMPERES 

Fig.  2O8 


due  to  foucault  currents  at  any  moment  is  equal  to  the 
corresponding  instantaneous  value  of  the  foucault  cur- 
rent loss  divided  by  the  instantaneous  value  of  the 
primary  pressure.  The  foucault  current  loss  may  be 
represented  by  the  symmetrical  cycle  shown  in  Fig. 
208,  where  the  ordinates  and  abscissas  are  respectively 
proportional  to  the  magnetism  and  its  rate  of  change. 


EFFECT   OF   COPPER   AND    IRON    LOSSES.       457 

The  area  of  this  cyclic  curve  is  proportional  to  the  total 
foucault  current  loss.  Its  effect  on  the  total  core  loss 
may  be  shown  by  plotting  a  cycle  having  abscissas 
equal  to  the  arithmetical  sum  of  the  corresponding 
abscissas  of  the  hysteresis  and  eddy  cycles  (Fig.  209). 
Finally,  the  total  transformer  exciting  current  may  be 
plotted  from  this  as  shown  by  C'  in  Fig.  207.  It  is 


Figf.  21O 

evident  that  the  hysteresis  and  eddy  components  of  the 
exciting  current  add  directly  together,  giving  the  total 
active  component  Cw,  which  is  nearly  sinusoidal  when 
the  impressed  pressure  is  sinusoidal.  If  the  foucault 
current  cycle  has  a  large  area,  its  effect  may  cause  a 
backwards  displacement  of  the  maximum  point  in  the 
exciting  current. 

When  the  secondary  circuit  is  closed,  the  form  of  the 


458 


ALTERNATING   CURRENTS. 


primary  current  is  changed  by  the  effect  of  the  sec- 
ondary current.  In  Fig.  210,  curve  E!  represents  the 
primary  pressure  curve,  n' C'  represents  the  exciting 
current  times  the  primary  turns.  Now,  if  by  closing 
the  secondary  circuit  through  the  non-inductive  resist- 
ance, a  current  C"  is  caused  to  flow,  and  its  ampere- 
turns  may  be  represented  by  the  curve  nlfC",  the 


Fig.  211 

effect  of  the  current  flowing  in  the  secondary  is  to  cause 
a  corresponding  increase  in  the  current  flowing  in  the 
primary  (Sect.  114),  and  the  ampere-turns  of  this  in- 
crease may  be  represented  by  curve  n'Cv  The  total 
primary  wave  is  similar  to  curve  nfCf,  which  is  the 
sum  of  n'Ci  and  n'Clf  and  the  primary  current  may 
be  directly  shown  from  this  curve  by  a  simple  change 
of  scale.  It  is  thus  shown  by  these  figures  that  the 


EFFECT  OF  COPPER  AND  IRON  LOSSES.   459 


— 

rW 

Xir 

oo~ 

138  ~»  per  Second. 
Primary  E.M.F.,  1030. 
Secondary  E.M.F.,  54.5. 
Secondary  Current  %/w 

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13 

12 
11 
10 
DC 
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X). 

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yOLTS  PRIMARY 

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300. 
llOOr 

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OPEN  CIRCUIT. 

Fig.  212 


SECONDARY  CLOSED  THROUGH  1O  LAMPS. 


/^ 

x^ 

L 

i 

1  1 

| 

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12<X). 

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'132—'    per  Second. 
Primary  E.M.F.,  1030. 
Secondary  E.M.F.,  49.3. 
Secondary  Current,  10.65. 

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1100. 

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FULL  LOAD. 

Fig-.  213 


460  ALTERNATING   CURRENTS. 

secondary  current,  when  in  phase  with  the  secondary 
pressure,  tends  to  reduce  the  distortion  and  lag  of  the 
primary  current,  exactly  as  has  already  been  proved 
analytically  (Sect.  114).  If  the  secondary  circuit  were 
inductive,  the  effect  would  be  altered  so  that  the  lag 
of  the  primary  current  would  be  larger,  as  shown  in 
Fig.  211,  and  as  has  already  been  proved  (Sects.  117 
and  1 1 8).  Figures  212  and  213  show  transformer 
curves  experimentally  observed  by  Professor  Ryan,* 
and  which  show  a  striking  resemblance  to  the  hypo- 
thetical curves  built  up  from  the  loss  cycles. 

*  Trans.  Amer.  Inst.  E.  £.,  Vol.  7,  p.  71. 


EFFICIENCY   AND    LOSSES.  461 


CHAPTER    XI. 

EFFICIENCY    AND    LOSSES     IN     TRANSFORMERS. 

121.  Transformer  Core  Losses  and  Magnetic  Densi- 
ties.—  The  commercial  efficiency  of  a  transformer  is 
the  ratio  of  the  electrical  output  of  the  secondary  coil 
to  the  corresponding  power  absorbed  by  the  primary 
coil.  It  may  be  written 

W"  =       W" 
W  ~  IV"  +  L 

where  W'  and  IV"  are  the  power  absorbed  and  de- 
livered respectively  by  the  primary  and  secondary  coils, 
and  L  is  the  total  loss  in  the  transformer.  This  total 
loss  is  made  up  of  the  C*R  losses  in  the  primary  and 
secondary  coils  and  the  losses  due  to  hysteresis  and 
foucault  currents  in  the  core.  The  C2R  loss  in  the 
secondary  winding  is  directly  proportional  to  the  square 
of  the  load  (secondary  output),  while  the  C2R  loss  in 
the  primary  is  nearly  proportional  to  the  square  of  the 
load,  though  it  contains  a  small  approximately  con- 
stant term  due  to  the  exciting  current  (Fig.  214).  The 
hysteresis  and  foucault  current  losses,  which  together 
constitute  the  Iron  Losses  or  Core  Losses,  have  been 


462 


ALTERNATING   CURRENTS. 


shown  experimentally  to  be  independent  of  the  load.* 
The  hysteresis  loss  is  directly  proportional  to  the  fre- 
quency and  approximately  proportional  within  the  limits 
of  magnetic  density  used  in  practice  to  the  1.55  or  1.6 
power  of  the  magnetic  density.  The  foucault  current 
loss  is  proportional  to  the  square  of  both  the  frequency 
and  the  magnetic  density.  Consequently,  for  fixed 
values  of  the  iron  losses  in  transformers  designed  for 
use  with  different  frequencies,  the  magnetic  density 
should  vary  inversely  with  some  power  of  the  fre- 


2100   3000   3600   1300 
WATTS 


5400   COOO   6600   7200 


Fig.  214 


quency  between  one  and  a  half  and  two.  The  table  in 
Section  97  gives  satisfactory  values  of  magnetic  densi- 
ties to  be  used  in  transformers  for  various  frequencies, 
though  the  values  there  given  are  commonly  exceeded 
in  American  transformers. 

The  actual  magnetic  densities  aimed  at  in  recent  trans- 
formers of  one  large  maker  may  be  represented,  for  a 
frequency  of  133,  by  the  following  formulas  and  the  first 
curve  shown  in  Fig.  215.  Where  the  output  is  below 


1500  watts, 


-7y-      r 


333°>  and  when  the  watts 


*  Ewing,  The  Dissipation  of  Energy  through  Reversals  of  Magnetism  in 
the  Core  of  a  Transformer,  London  Electrician,  Vol.  28,  p.  1  1  1  ;  Ewing  and 
Klaassen,  Magnetic  Qualities  of  Iron,  London  Electrician,  Vol.  32,  p.  713. 


EFFICIENCY   AND   LOSSES.  463 

output  is  above  1500  watts,  ^max  =  35OO.  These  trans- 
formers may  be  used  on  circuits  having  frequencies 
from  60  to  135,  in  which  case  magnetic  densities  are 
inversely  proportional  to  the  frequencies.  In  older 
transformers,  where  poorer  iron  was  used,  the  densities 
aimed  at  by  the  same  maker  are  shown  by  the  second 
curve  of  Fig.  215. 

The  magnetic  densities  in  the  transformers  of  another 
large  manufacturer  lie  between  3600  and  2800  at  a 
frequency  of  125,  in  transformers  of  capacities  between 
500  and  30,000  watts.  Similar  magnetic  densities  are 
aimed  at  in  the  transformers  of  a  third  large  manu- 
facturer, and  all  successful  American  manufacturers 
keep  pretty  closely  within  these  limits. 

The  percentage  which  the  core  losses  bear  to  the  out- 
put in  well-designed  transformers  varies  greatly.  The 
average  for  transformers  not  smaller  than  6  K.W. 
capacity  and  not  larger  than  20  K.W.  capacity,  may 
be  said  to  range  between  f  and  2  per  cent.  For  smaller 
transformers,  this  percentage  increases.  For  3  K.W. 
transformers  2  per  cent  is  a  fair  value,  though  3  per 
cent  is  exceeded  in  some  transformers  of  this  size ;  and 
for  500  watt  transformers  5  per  cent  is  not  bad. 

First  class  transformers  should  have  core  losses  not 
exceeding  the  following:  i  K.W.,  30  watts;  ij  K.W., 
40  watts ;  2  K.W.,  50  watts ;  2j  K.W.,  60  watts ; 
4  K.W.,  80  watts;  6|-  K.W.,  100  watts;  i;J  K.W., 
150  watts.  Intermediate  sizes  will  have  proportional 
losses.  In  some  transformers  these  figures  are  bettered 
on  the  higher  commercial  frequencies,  as  in  the  case  of 
two  of  7500  watts  capacity,  built  by  different  makers, 


464 


ALTERNATING   CURRENTS. 


•xvw  -a 


EFFICIENCY   AND    LOSSES. 


465 


in  which  the  core  losses  varied  from  75  to  125  watts 
depending  on  the  frequency,  the  test  frequencies  being 
60  and  125.* 

The  following  table  of  the  exciting  currents  of  good 
transformers  is  taken  from  the  results  of  numerous 
tests  by  Professor  Ryan.f 


Capacity. 

Exciting  Current. 

Approximate  Per  Cent  of 
Primary  Full  Load  Current. 

250 

.040  Ampere 

14. 

500 

.050        " 

7.2 

IOOO 

•055 

5-o 

2OOO 

.080        " 

3-8 

6500 

.100 

i-5 

17500 

.200 

i.i 

The  data  of  this  table  were  gained  from  tests  of 
transformers  of  various  makers  designed  for  a  primary 
pressure  of  1000  volts  at  a  frequency  of  133,  but  are 
rather  high  for  the  better  grade  of  transformers. 

122,  Copper  Losses. — The  C2R  loss  in  transformers 
is  ordinarily  between  ij  per  cent  and  3-^  per  cent. 
This  is  divided  with  approximate  equality  between  the 
primary  and  secondary  windings.  Sometimes  this  loss 
is  permitted  to  reach  5  per  cent,  but  in  the  better  trans- 
formers it  is  more  often  between  2  per  cent  and  3  per 
cent.  The  primary  and  secondary  coils  of  good  com- 
mercial transformers  of  later  design  are  so  disposed 

*  Bull.  Univ.  of  Wisconsin,  Vol.  I,  No.  II;  Jackson,  Electrical 
Journal,  Vol.  I,  p.  78,  and  N.  Y.  Elect.  Engineer,  Vol.  20,  p.  183. 

t  Ryan,  The  Efficiency  of  Alternating  Plants,  N.  Y.  Elect.  Engineer, 
Vol.  13,  p.  12. 


2H 


466         ALTERNATING  CURRENTS. 

that  the  magnetic  leakage  is  practically  negligible, 
though  in  earlier  transformers  this  was  not  true.*  Con- 
sequently, the  change  in  the  ratio  of  transformation 
causing  a  drop  in  the  secondary  pressure  as  the  load 
increases,  is  practically  all  caused  by  the  copper  losses. 
If  the  total  C2R  loss  is  equally  divided  between  the 
primary  and  secondary  windings,  and  magnetic  leakage 
is  negligible,  the  following  relations  exist  for  trans- 
formers having  a  magnetic  circuit  wholly  of  iron  : 


and,  approximately, 


m*# 

jfV 

^~ 

~~Tr  ~    > 
nn 

Z77" 

M=VLrLn 

_  L' 

~~  k  = 

_  _ 

C"     k     C" 


C  ' 
where  k  is  the  ratio  of  transformation.     —^  is  usually 

T 

very  small  compared  with  — 

A- 

123.   Rise  of  Temperature  and  Radiating  Surface.  — 

The  windings  of  transformers  are  usually  embedded 
largely  in  the  iron  core,  and  the  whole  transformer  is 
enclosed  in  a  water-proof  iron  case;  and  their  rise  of 
temperature  is  as  much  due  to  the  heating  of  the 
core  by  core  losses  as  to  the  copper  losses.  If  trans- 
formers were  placed  in  the  open  air,  the  entire  external 
surface  could  be  assumed  to  be  effective  in  dissipating 
heat  by  radiation  and  convection,  but  on  account  of 

*  Ryan,  Irans.  Amer.  hist.  E.  E.,  Vol.  7,  p.  n. 


EFFICIENCY   AND    LOSSES.  467 

the  enclosing  case,  convection  from  the  surface  can- 
not take  place,  and  all  the  heat  must  be  radiated  to  the 
wall  of  the  case,  or  conducted  thereto  through  the  poor 
heat  conductors  which  are  used  to  electrically  insulate 
the  transformer  from  its  case.  The  conditions  there- 
fore point  to  the  conclusion  that  for  a  given  liberation 
of  heat  per  square  centimeter  of  surface,  the  tempera- 
ture is  likely  to  be  higher  in  transformers  than  in 
dynamo  fields.  On  the  other  hand,  transformer  coils 
may  always  be  designed  to  be  lathe  wound,  and  there- 
fore may  be  more  effectually  insulated  than  dynamo 
field  coils  ;  and  as  space  is  not  so  valuable  in  transfor- 
mers, more  liberal  use  may  be  made  of  mica,  varnished 
canvas,  fibre,  and  wood.  It  is  therefore  possible  to 
safely  run  transformers  with  the  windings  at  a  consid- 
erably higher  temperature  than  dynamos,  and  60°  Centi- 
grade (108°  F.)  may  be  set  as  a  safe  limit  to  the  rise  in 
temperature.  A  high  temperature  limit  has  a  marked 
disadvantage  in  causing  an  undue  drop  in  pressure  as 
the  transformer  heats  up,  by  increasing  the  resistance 
of  the  windings,  and  while  many  transformers  exceed 
the  temperature  limit  named,  many  of  the  best  types 
avoid  the  difficulty  from  drop  in  the  pressure  by  not 
exceeding  40°  rise.  As  the  rise  in  temperature  also 
increases  the  electrical  resistance  of  the  iron  core,  it 
decreases  the  foucault  current  Joss,  so  that,  as  sug- 
gested by  Elihu  Thomson,  it  would  be  advantageous 
to  have  the  core  of  a  transformer  operated  at  a  high 
temperature  while  the  windings  were  kept  cool.  This 
cannot  be  conveniently  arranged  in  small  transfor- 
mers, but  the  cooling  of  the  conductors  of  very  large 


468  ALTERNAThNG  CURRENTS. 

transformers  has  been  effected  by  making  the  con- 
ductors tubular  and  passing  a  cool  liquid  through  them. 

It  is  practically  impossible  to  fix  any  averages  for  the 
external  surface  of  transformers  per  watt  lost  in  the 
core  and  windings,  on  account  of  the  very  varied 
arrangements  of  the  coils  with  reference  to  the  core, 
and  the  effect  of  the  containing  case.  For  small  and 
medium  transformers  it  is  usual  to  make  the  design  as 
compact  as  possible,  and  no  particular  trouble  from 
heating  is  experienced  if  the  losses  are  not  excessive, 
since  the  losses  ought  to  be  quite  small  In  large  trans- 
formers the  same  plan  may  be  adopted,  and  some  device 
may  be  arranged  for  cooling  the  conductors,  such  as  cir- 
culating a  liquid  through  them  or  blowing  air  through 
ducts  in  the  core.  Figure  215  a  shows  the  rise  of  tem- 
perature of  the  conductors  as  a  function  of  the  period  of 
operation  of  a  200  K.W.  air-blast  transformer  with  and 
without  the  blast.  Point  D  shows  the  temperature  of 
the  discs  after  the  transformer  had  been  operated  seven 
hours  with  blast  on  ;  curve  A  shows  the  temperature  of 
the  windings  when  operated  at  full  load  without  blast ; 
curve  B  shows  the  temperature  of  the  windings  when 
operated  at  full  load  with  blast  of  1040  cubic  feet  per 
minute  ;  and  curve  C  shows  the  temperature  of  the  air 
issuing  from  the  transformer.  The  core  and  windings  of 
some  transformers  are  immersed  in  oil,  which  fills  the  case, 
so  as  to  give  a  better  opportunity  for  the  heat  to  escape. 

124.  Current  Density  in  Transformer  Conductors.  — 
The  current  density  in  transformer  windings  varies  be- 
tween 1000  and  2000  circular  mils  per  ampere.  It  is 
not  unusual  to  make  the  density  somewhat  smaller  in 


EFFICIENCY    AND    LOSSES. 


469 


the  secondary  than  in  the  primary,  and  the  values  in 
the  best  transformers  frequently  fall  between  1000  and 
1 500  circular  mils  per  ampere  for  the  primary  coil,  and 
between  1200  and  2000  for  the  secondary  coil.  On  the 
other  hand,  some  designers  make  the  density  of  current 
greater  in  the  secondary  windings,  while  others  make 
the  density  about  the  same  in  each.  As  the  primary 


90 
£«> 

ui 

S" 

III 


Q 

z    50 

u 

tr 
=    40 

D: 
a    30 

LU 

*•    20 


10 


c 


1S345G78910 

TIME  IN  HOURS. 

Fig.  215  a 

wire  of  nearly  all  commercial  transformers  is  much 
smaller  than  the  secondary  conductor,  insulation  takes 
up  much  more  space  in  the  primary  coil,  and  this  coil 
occupies  more  space  than  the  secondary  coil  unless  the 
primary  current  density  is  considerably  greater. 

125.  Testing  Transformers. — Methods  for  testing  the 
efficiency  of  transformers  and  for  determining  their 
core  losses  have  received  a  large  amount  of  attention. 


470         ALTERNATING  CURRENTS. 

The  more  important  methods  are  explained  in  the  fol- 
lowing pages. 

Transformers  are  ordinarily  worked  on  loads  com- 
posed of  incandescent  lamps,  which  are  practically 
non-inductive.  Consequently  there  is  no  difficulty  in 
determining  the  power  in  the  secondary  circuit,  since 
the  indications  of  a  proper  amperemeter  and  voltmeter 
are  sufficient.  The  power  in  watts  is  given  by  the 
product  of  the  indications,  since  the  current  and  press- 
ure agree  in  phase ;  a  load  composed  of  a  liquid  resist- 
ance serves  equally  well.  If  the  load  is  composed  of  a 
non-inductive  wire  resistance  which  is  not  appreciably 
heated  by  the  current  delivered  by  the  transformer,  the 
readings  of  the  amperemeter  and  voltmeter  may  be 
checked  by  comparing  their  indications  with  the  meas- 
ured values  of  the  resistance. 

The  measurement  of  the  power  absorbed  by  the  pri- 
mary is  not  so  simple,  since  the  current  and  pressure  do 
not  agree  in  phase.  The  same  is  true  of  the  measure- 
ments in  the  secondary  circuit  if  the  load  is  reactive. 

i.  Ryaris  and  Merritfs  Method.  —  One  of  the  earliest 
thorough  tests  on  commercial  transformers  was  that 
carried  out  by  Professors  Ryan  and  Merritt  in  1889.* 

In  this  series  of  tests  the  curves  of  current  and 
electric  pressure  were  determined  by  Ryan's  method 
(Sect.  78,  4),  and  the  effective  values  of  these  quantities 
and  of  the  power  in  the  circuit  were  determined  from  the 
curves,  by  the  first  method  given  in  Section  81,  except  in 
the  case  of  the  secondary  current,  which  was  directly 
measured  by  an  amperemeter.  The  connections  are 

*  Trans.  Amer.  Inst.  E.  E.,  Vol.  7,  p.  i;   Elect.   World,  Vol.  14,  p.  419. 


EFFICIENCY   AND   LOSSES. 


471 


shown  in  Fig.  216;  T  is  the  transformer,  MM  are  the 
primary  leads  to  the  transformer,  LL  are  incandescent 
lamps  connected  as  load  to  the  secondary,  KK  is  a  non- 
inductive  resistance  connected  across  the  primary  leads, 
/  is  an  amperemeter,  A  is  a  contact  maker,  E  is  a  Ryan 
electrometer  with  accompanying  devices,  and  GG  is  a 
series  of  switches.  The  object  of  these  switches  is  to 


M 


v  K  K  K 

uiknjuuuuuuuuuuu^ 


Fig.  216 

connect  the  contact  maker  alternately  with  different  parts 
of  the  circuit,  so  that  the  curves  of  primary  and  second- 
ary pressure  and  primary  current  may  be  traced. 

The  curve  of  pressure  at  the  terminals  of  the  non- 
inductive  resistance  R  is  evidently  the  same,  if  taken 
to  a  proper  scale,  as  the  curve  of  secondary  current. 
In  order  to  avoid  handling  the  large  primary  pressure 
at  the  contact  maker  and  electrometer,  the  calibrated 
resistance  KK  is  used.  The  curve  of  pressure  taken 


4/2 


ALTERNATING   CURRENTS. 


between  any  two  points  on  this  resistance,  when  given 
a  proper  scale,  is  evidently  the  same  as  the  curve  of 
total  pressure.  For  the  non-inductive  resistances  R 
and  KK,  Ryan  and  Merritt  used  incandescent  lamps 
which  they  had  "rated." 

These  tests  were  carried  out  on  a  ten-light  (500  watt) 
transformer  when  operated  with  the  secondary  circuit 


138  ~  PER  SECOND 
PRIMARY  E.  M.F.   1030 
SECONDARY  E.M.F.  52.3 
SECONDARY  CURRENT   1.26 


Pig.  217 

open,  and  with  the  secondary  loaded  respectively  with 
one  lamp,  with  five  lamps,  and  with  ten  lamps.  The 
curves  gained  under  these  conditions  are  given  in  Figs. 
212,  217,  218,  and  213. 

In  addition  to  the  determination  of  the  curves  of 
current  and  pressure,  the  various  quantities  were  also 
directly  measured.  The  effective  values  deduced  from 
the  curves  agree  quite  closely  with  those  directly  deter- 


EFFICIENCY   AND   LOSSES. 


473 


mined,  but  are  of  no  use  in  determining  the  perform- 
ance of  the  transformer,  unless  the  phase  differences 
of  pressures  and  currents  be  determined  in  some  man- 
ner. This  is  practically  done  by  using  the  curves. 


PER  SECOND 
PRIMARY  E.  M.F.  1040 
SECONDARY  E.  M.F.  51.0 
SECONDARY  CURRENT  5. 


Fig.  218 


The  results  given  by  these  tests  reduced  to  a  uniform 
primary  pressure  of  1020  volts  are  as  follows  : 


No  Load. 

%  Load. 

^i/2  Load. 

Full  Load. 

Secondary  pressure  

C2.  ~\ 

">2.  'I 

CO.  I 

47.  c 

Watts  absorbed  by  primary  
Watts  delivered  by  secondary  
Commercial  efficiency 

96.1 
0.0 

o  o 

I59-I 
64-3 
41.  1 

388.6 
300.9 

77.  c 

607.9 

525-° 
86.6 

Total  loss  in  watts 

06   I 

Q4.8 

87.7 

82.0 

C^R  loss  in  primary 

O  4 

O  Q 

•}.  -} 

8.7 

C^R  loss  in  secondary  

o.o 

O.O 

1.  3 

4.c 

Core  losses  .  ,  

QC.7 

0-2.0 

83.1 

69.7 

474 


ALTERNATING   CURRENTS. 


Figure  219  gives  the  curves  obtained  by  taking  the 
products  of  the  instantaneous  pressures  and  currents 
through  one-half  a  period.  The  proportions  of  nega- 
tive work  due  to  the  wattless  component  of  the  current 
to  positive  work  in  the  different  cases  are  :  open  cir- 
cuit, 6.8  per  cent ;  one-tenth  load,  3.9  per  cent ;  one- 
half  load,  .96  per  cent;  full  load,  .36  per  cent.  The 


Fig.  219 

power  factors  in  the  different  cases  are,  therefore,  86.4 
per  cent,  92.2  per  cent,  98.1  per  cent,  and  99.3  per  cent. 
The  curves  of  pressure  and  current,  which  are  here 
shown,  give  in  an  interesting  manner  the  effect  of  the 
secondary  current  on  the  form  of  the  primary  current 
wave.  The  curves  also  show  the  effect  of  magnetic 
leakage  in  retarding  the  phase  of  the  secondary  press- 
ure in  relation  to  that  of  the  primary  pressure.  A 


EFFICIENCY    AND   LOSSES. 


475 


cross-section  of  the  transformer  is  given  in  Fig.  220, 
which  shows  the  arrangement  of  the  primary  and  sec- 
ondary coils.  In  this  transformer  the  magnetic  leakage 
amounted  to  1.2  per  cent  on  open  circuit  and  increased 
with  the  load  to  as  much  as  5.4  per  cent  on  full  load. 
At  the  same  time  the  secondary  pressure  wave  fell  back 
from  exact  opposition  to  the  primary  pressure  wave 
(180°  lag)  to  nearly  190°  lag. 

In  Fig.  218^  are  given  the  curves  of  a  Stanley  trans- 
former with  a  capacity  of  17500  watts,  when  the  second- 


Fig-.  220 

ary  circuit  is  open.  This  transformer  was  tested  by 
Professor  Ryan  in  1892,*  with  the  following  reduced 
results : 

Primary  pressure,  1000  volts  ;  secondary  pressure,  no 
load,  50.80  volts;  full  load,  49.67  volts;  drop,  1.16  volts 
or  2.32  per  cent;  efficiency  at  full  load,  96.9  per  cent; 
at  half  load,  97.1  per  cent;  one-quarter  load,  96.0  per 
cent;  and  one-eighth  load,  93.0  per  cent;  exciting 
current,  .195  amperes;  power  absorbed  with  second- 
ary open  (core  losses),  137  watts;  magnetic  leakage 


*  N.  Y.  Elect.  Engineer,  Vol.  14,  p.  298. 


4/6 


ALTERNATING   CURRENTS. 


practically  negligible.  The  frequency  on  which  the 
transformer  was  tested  was  practically  133.  The  pro- 
portionately small  value  of  the  magnetizing  component 


Fig.  218y2 

of  the  exciting  current  in  this  transformer  (the  power 
factor  shown  by  the  results  marked  on  the  figure  is  .86) 
decreases  the  distortion  of  the  curve  of  exciting  current 


EFFICIENCY   AND    LOSSES.  477 

and  causes  its  maximum  point  to  be  displaced  to  a  posi- 
tion in  advance  of  the  position  of  zero  pressure  (Sect.  1 20). 

2.  H op  kins  on  s  Method,  In  the  preceding  method 
the  losses  and  efficiencies  are  determined  by  taking  dif- 
ferences between  measured  quantities  which  are  of  con- 
siderable magnitude,  each  of  which  may  be  affected  by 
considerable  errors  of  observation,  which  errors  enter 
directly  into  the  value  of  the  difference.  The  differ- 
ences thus  obtained  are  therefore  not  reliable. 

Dr.  John  Hopkinson  modified  the  method  by  using 
two  similar  transformers,  connected  (Fig.  221)  so  that 
one  transformed  up  the  pressure  supplied  to  its  thick 
wire  coil,  and  the  other  transformed  this  pressure  down 
again.  The  respective  pressures  and  currents  in  the 
low-pressure  coils  of  the  two  transformers  when  thus 
arranged  are  therefore  nearly  equal.  By  measuring 
their  differences  and  determining  their  phase  relations, 
the  losses  in  the  two  transformers  are  obtained.  Assum- 
ing the  two  transformers  to  be  similar,  one-half  the  total 
loss  is  due  to  each.  The  efficiency  is  then  obtained 
from  this  loss  measurement  and  a  measurement  of  the 
current  and  pressure  in  the  secondary  circuit  of  the 
second  transformer,  which  consists  of  a  non-inductive 
resistance.  To  determine  the  differences  and  their 
phase  relations,  a  contact  maker  and  electrometer  were 
used  (Sect.  78,  3). 

The  connections  were  arranged  as  shown  (Fig.  221)* 
and  a  series  of  curves  were  obtained  for  various  loads 
which  are  shown  in  Fig.  222  (a,  by  c,  and  d). 

*  Hopkinson's  Dynamo  Machinery  and  Allied  Subjects,  p.  187;  Elect. 
World,  Vol.  20,  p.  40. 


478  ALTERNATING   CURRENTS. 


No,? 


No,  2 


TO  RESISTANCE 


TO  CONTACT  MAKER 


No,1 


No.  2 


EFFICIENCY   AND   LOSSES. 


479 


The  results  of  these  tests  showed  the  efficiencies  of 
Westinghouse  transformers  of  6500  watts  capacity  to  be 


238  270 

TIME 

Figr.  222  a 


282 


96.9  per  cent  at  full  load,  96  per  cent  at  half  load,  and 
92  per  cent  at  quarter  load ;  the  drop  of  pressure 
between  no  load  and  full  load  to  be  between  2.2  and  3 


4 1 


per  cent;    the  core  losses  in   each   transformer  to  be 
about   114  watts;   the  magnetic  leakage  in  one  trans- 


480 


ALTERNATING   CURRENTS. 


former  appeared  to  be  small,  but  in  the  other  to  be  con- 
siderable. 

3.  Mordeys  Method.  Mordey  suggested  the  follow- 
ing method  for  determining  the  losses  in  a  transformer 
at  any  desired  load:*  The  transformer  to  be  tested  is 
worked  at  its  normal  load  until  a  constant  temperature 
is  reached  which  is  determined  by  a  thermometer.  A 


I10 

5 
20 


30 


284 


100 


0      ° 


Fig.  222  c 


continuous  current  is  then  passed  through  the  windings 
of  such  a  magnitude  that  the  C2R  loss  from  this  cur- 
rent is  sufficient  to  maintain  the  temperature  of  the 
transformer.  The  continuous  current  C2R  loss  is  then 
equal  to  the  total  losses  with  the  alternating  current. 
The  continuous  current  C^R  loss  is  readily  determined 
by  voltmeter  and  amperemeter,  or  by  wattmeter.  The 


*  Alternate  Current  Working,  Jour.  Inst.  E.  E.,  Vol.  18,  p.  608. 


EFFICIENCY   AND   LOSSES. 


481 


tests  by  this   method  are  laborious  and  impractical  on 
account  of  the  time  required. 

4.  Calorimetric  Method.  This  method  has  been  used 
by  many  experimenters.*  The  transformer  is  taken 
from  its  iron  case  and  sealed  up  in  a  tin  or  copper  case 
which  is  immersed  in  a  water  calorimeter  (Fig.  223). 
The  loss  in  the  transformer  at  any  desired  load  on  the 


ff.10 

5 
20 


258  270 

TIME 

Fig.  222  d 


294 


secondary  is  determined  from  the  rate  at  which  the 
water  in  the  calorimeter  rises  in  temperature,  provided 
the  heat  absorbed  by  the  transformer  and  calorimeter 
(water  equivalent),  and  the  rate  of  loss  of  heat  from  the 
sides  of  the  calorimeter,  have  been  determined,  so  that 
a  proper  correction  may  be  applied  to  the  results.  The 
determination  of  the  elements  entering  into  this  correc- 

*  Duncan,    Electrical    World,   Vol.   9}   p.    188 ;   Roiti,    La   Lumiere 
Electrique,  Vol.  35,  p.  528. 
21 


482 


ALTERNATING    CURRENTS. 


tion  is  likely  to  introduce  considerable  errors.     These 
are  decreased  somewhat  by  using  oil  in  the  calorimeter, 

when  the  transformer  may 
be  directly  immersed  with- 
out the  enclosing  case.  The 
errors  due  to  the  heat  ab- 
sorbed by  the  transformer 
and  calorimeter  may  be  elim- 
inated by  arranging  the  ap- 
paratus so  that  a  slow  con- 
stant flow  of  water  is  passed 
through  the  calorimeter  (Fig. 
224).  The  entering  water  should  have  a  constant 
temperature.  If  the  flow  of  water  is  of  constant 
volume,  and  the  transformer  has  attained  a  con- 
stant temperature,  there  will  be  a  constant  difference 
of  temperature  between  the  water  at  entrance  and 


LJ 


U 


Fig.  223 


Fig.  224 

exit.  This  may  be  measured  by  thermometers.  From 
this  difference  of  temperature  and  the  weight  of  water 
flowing  per  minute  through  the  calorimeter,  the  energy 
expended  in  the  transformer  and  given  up  to  the  water 


EFFICIENCY   AND   LOSSES.  483 

may  be  deduced.  A  correction  must  be  applied  on 
account  of  the  loss  of  heat  due  to  radiation  from  the 
calorimeter.  A  more  satisfactory  arrangement  of  the 
calorimeter  test  is  to  place  the  hermetically  sealed 
transformer  in  a  vessel  with  ice  packed  around  it.  If 
the  vessel  is  properly  protected  from  external  heating 
effects,  the  melting  of  the  ice  will  practically  all  be  due 
to  heat  from  the  transformer,  and  the  losses  in  the 
transformer  may  be  determined  from  the  amount 
melted  in  a  given  time.  A  correction  must  be  applied 
for  the  melting  before  the  transformer  has  reached  a 
constant  temperature. 

It  is  difficult,  at  best,  to  get  very  satisfactory  results 
from  the  calorimetric  methods,  on  account  of  the  difficul- 
ties inherent  in  the  exact  determination  of  temperatures 
or  of  heat  losses.  The  best  method  of  work  is  prob- 
ably to  directly  calibrate  the  calorimeter  by  inserting  in 
the  place  of  the  transformer  a  known  resistance  and 
passing  through  it  such  a  continuous  current  as  will 
give  the  same  heating  effect  as  the  transformer.  Then 
the  energy  absorbed  by  the  resistance  is  equal  to  the 
transformer  losses. 

5.  Split  Dynamometer  Method.  The  split  dynamome- 
ter may  be  used  for  directly  determining  the  power 
absorbed  by  a  loaded  transformer  if  the  magnetic  leak- 
age is  negligible.  In  this  case  an  electrodynamometer 
is  connected  with  one  coil  in  each  circuit.  Then,  practi- 
cally, e'  =  R'c'  +  (^7)*"-  But  e"  =  c"  (R  +  R"),  where 

e1,  e" ;  c\  c"  ;  n' ,  u"  ;  and  R',  R"  are  respectively  the 
pressures,  currents,  turns,  and  resistances  in  the  pri- 


484  ALTERNATING   CURRENTS. 

mary  and   secondary  coils,  and  where  R  is  the  resist- 
ance of  the  external  secondary  circuit.     Then 

e'c'  =  R'c'2  +  -n  (R  +  R")c'c". 

The  energy  absorbed  by  the  primary  circuit  is,  there- 
fore, 


< 

Tn" 

The  integral  of  the  first  term  of  the  right-hand  mem- 
ber is  equal  to  the  square  of  the  primary  current,  and 
that  of  the  second  term  is  C'C"  cos  <f>  =  kDy  where  D  is 
the  reading  of  the  split  dynamometer  and  k  is  its 
constant  (Sect.  44,  5).  Consequently, 

W  =  C'2R'  +  ^  (R  +  R")  kD. 

As  already  said,  this  method  is  only  accurate  when 
magnetic  leakage  is  negligible. 

6.  Voltmeter  and  Ammeter  Methods.     The  power  in 
the  primary  and  secondary  circuits  may  be  measured 
by  any  of  the  methods  given  in  Section  44  for  measur- 
ing the  power  in  an  alternating-current  circuit  by  means 
of  voltmeters  and  amperemeters.     The  numerical  effi- 
ciency will  be  found  from   the  ratio  of  the   secondary 
and  primary  energies,  and  the  transformer  losses  from 
their  difference. 

7.  Wattmeter   Method.       Where    satisfactory    watt- 
meters   are   at    hand,   it    is    evident    that   one    may  be 
placed  in  the  primary  circuit  of   a  loaded  transformer 
and  another  in  the  secondary  circuit.     Then  the  ratio  of 
their  readings  is  the  efficiency  of  the  transformer.     Or, 


EFFICIENCY   AND   LOSSES.  485 

a  wattmeter  may  be  used  in  the  primary  circuit  and  an 
amperemeter  and  voltmeter  may  be  used  to  determine 
the  output,  if  the  transformer  is  worked  on  a  non-induc- 
tive load.  The  wattmeter  was  used  by  Fleming  in  a 
very  extended  series  of  transformer  tests,  and  found  to 
be  more  satisfactory  than  either  of  the  methods  in 
which  amperemeters  and  voltmeters  are  used  to  deter- 
mine the  energy  absorbed  by  the  primary  circuit.* 

8.  Stray  Power  Methods.  A  very  convenient  method 
of  measuring  the  efficiency  of  transformers  is  to  de- 
termine the  various  losses  directly,  and  thence  the 
efficiency.  The  iron  losses  may  be  determined  by 
measuring  with  a  wattmeter  the  power  absorbed  by  the 
transformer  when  the  secondary  circuit  is  open.  The 
copper  losses  for  any  load  are  readily  calculated  when 
the  secondary  and  exciting  currents  and  the  primary 
and  secondary  resistances  are  known.  The  exciting 
current  may  be  measured  at  the  same  time  that  the 
iron  losses  are  determined,  and  the  resistances  may  be 
measured  by  a  bridge.  For  a  given  load,  the  secondary 
current  is  a  fixed  quantity.  The  efficiency  is  then, 

practically, 

W" 


where  /  represents  the  measured  iron  losses  and  k  the 
ratio  of  transformation. 

8  a.  A  still  more  convenient  method,  which  may  be 
readily  used  in  central  stations  for  testing  transformers, 
is  to  measure  the  iron  losses  by  a  wattmeter,  as  ex- 

*  your.  Inst.  £.  E.,  Vol.  21,  p.  623;  Elect.  World,  Vol.  20,  p.  413. 


486 


ALTERNATING   CURRENTS. 


plained  above.  The  copper  losses  may  then  be  meas- 
ured by  short-circuiting  the  secondary  through  an 
amperemeter,  and  adjusting  the  primary  pressure  until 
the  full  load  current,  or  any  desired  fraction  thereof, 
passes  through  the  amperemeter.  The  reading  of  a 
wattmeter  in  the  primary  circuit  is  now  nearly  equal 
to  the  copper  losses,  since  the  pressure  and  maximum 
magnetic  density  must  be  very  small,  and  the  iron  losses 
are  therefore  almost  or  entirely  negligible.  The  exact 
copper  losses  may  be  determined  by  measuring  and 


OPEN 
^      CIRCUIT 


Pig-.  225 

correcting  for  the  small  iron  loss.  The  tests  for  the 
iron  losses  may  be  most  conveniently  made  by  using 
the  low  resistance  coil  of  the  transformer  as  the  primary 
coil  (Fig.  225). 

This  method  of  testing  may  be  used  with  satisfaction 
where  numerous  transformers  must  be  tested,  since  the 
losses  and  efficiency  may  be  determined  expeditiously 
and  with  the  expenditure  of  little  power.  When  com- 
bined with  a  run  of  several  hours  with  full  load  current, 
the  secondary  circuit  being  made  up  of  impedance  coils, 
the  method  proves  very  economical  for  shop  tests.  The 


EFFICIENCY   AND    LOSSES. 


487 


method  was  adopted  by  Mr.  A.  H.  Ford  for  a  very  com- 
plete series  of  tests  of  American  transformers.* 

8  b.  Ayrton  and  Sumpner  have  devised  a  method 
not  so  serviceable  as  the  last,  but  still  quite  useful, 
in  which  two  transformers  of  the  same  size  and  make 
are  opposed  to  each  other.  The  method  of  connecting 


•Q- 


Pig-.  225  a 

up  is  as  in  Fig.  225  a,  in  which  A  and  B  are  the  trans- 
formers with  their  primaries  connected  in  relatively 
opposite  directions  to  the  leads  and  their  secondaries  in 
series.  The  pressures  of  the  secondaries  are  thus  op- 
posed. A  transformer  T  is  inserted  with  its  secondary 


*  Tests  of  American  Transformers,  Bull,  of  Univ.  of  Wisconsin,  Vol.  I, 
No.  ii. 


488         ALTERNATING  CURRENTS. 

in  circuit  with  the  secondaries  of  A  and  B.  By  varying 
the  resistance  R,  the  pressure  of  T  may  be  regulated  so 
that  any  desired  current  will  pass  through  the  second- 
aries of  A  and  B.  Then  the  output  of  T  measured  by  the 
wattmeter  W2  will  give  approximately  the  copper  losses 
of  A  and  B  plus  the  loss  in  leads  and  instruments. 
The  power  supplied  to  the  primaries  of  A  and  B  by  the 
alternator,  measured  by  the  wattmeter  W^,  gives  approx- 
imately the  iron  losses  of  the  two  transformers.  From 
the  data  thus  derived  the  efficiencies  may  be  obtained. 

126.  Iron   Losses    Independent   of    Load.  —  That   the 
value  of   the    iron    losses    is   independent  of   the   load 
carried     by     a     transformer,    was     first     conclusively 
proved    by   Ewing  (Sect.    127).     The    same   thing  has 
been  proved  by  Fleming's    experiments.       Figure  226 
is  plotted  from  one  of  his  tests  made  on  a  transformer 
of   4000   watts    capacity.     The    ordinates    of   line   AB 
represent  the  differences  of  the  power  in  the  primary 
and    secondary    circuits    as    measured    by    wattmeters. 
The  calculated   copper  losses    are    represented   by  the 
ordinates    of    the    line    CD.      The    difference    of    the 
ordinates  of  the  lines  AB  and  CD  at  any  point  is  the 
iron  loss  for  the  particular  load.     The  lines  AB  and  CD 
are  approximately  parallel,  which  shows  that  the  iron 
losses  are  practically  constant,  regardless  of   the  load. 
Therefore  the   stray  power  methods  of   testing   trans- 
formers give  efficiencies  which  are  entirely  reliable. 

127.  Ewing's  Experiment  showing  that  Iron   Losses 
are  Constant.  —  Ewing's  very  neat  plan  for  proving  this 
point  was  designed  to  get  at  the  matter  directly.     Two 
small   transformers   were    made   up   exactly   alike,  the 


EFFICIENCY   AND   LOSSES. 


489 


cores  of  which  consisted  of  insulated  iron  wire  wound 
into  the  form  of  a  ring.  Over  this  were  uniformly 
wound  two  layers  of  wire  making  a  primary  coil,  and 
another  two  layers  making  a  secondary  coil.  In  operat- 
ing, the  primary  and  secondary  coils  were  respectively 
connected  in  series,  but  the  two  halves  of  each  coil  in 
one  transformer  were  so  connected  as  to  be  in  magnetic 
opposition  (Fig.  227).  The  core  of  one  transformer  was 
therefore  magnetized  and  that  of  the  other  was  not. 
While  the  C^R  losses  at  any  load  were  equal  in  the  two, 


TOT|M 


LOSSES 


0         300       600       900      1200     1500      1800     2100      2400      2700      3000     3300      3COO      3900 
OUTPUT  IN  SECONDARY  WATTS. 

Fig-.  226 

the  transformer  with  magnetized  core  heated  more  when 
put  in  operation  than  the  other  transformer,  but  the 
temperature  of  the  second  was  brought  to  equality  with 
that  of  the  first  by  passing  a  continuous  current  through 
the  insulated  wire  of  the  core.  The  energy  expended 
in  heating  the  second  core  by  the  continuous  current 
was  thus  equal  to  that  expended  in  the  first  core  due 
to  iron  losses.  The  equality  of  temperature  was  deter- 
mined by  means  of  thermo-electric  couples  embedded  in 
the  cores,  which  were  connected  in  series  through  a 
galvanometer  (Fig.  227).  In  this  experiment  it  was 
found  that  after  a  balance  of  temperature  was  once 


490 


ALTERNATING    CURRENTS. 


obtained  it  was  unaltered  by  any  changes  in  the  loads 
of  the  transformers,  thus  showing  that  the  core  losses 
in  the  magnetized  transformer  were  independent  of 
the  load.* 

128.    Regulation   Tests.  —  The    regulation    of    trans- 
formers which  are  used   in   incandescent  lighting  is  a 


ALTERNATOR 


Fig.   227 

matter  of  much  moment,  and  regulation  tests  are  of 
almost  equal  importance  to  the  tests  of  losses  and  tem- 
peratures. The  ordinary  method  of  making  regulation 
tests  is  to  place  a  voltmeter  across  the  primary  circuit 
and  another  across  the  secondary  circuit  of  the  trans- 
former to  be  tested.  At  no  load,  the  reduced  readings 

*  Ewing  and  Klaassen,  Magnetic  Qualities  of  Iron,  London  lUectrician, 
Vol.  32,  p.  713. 


EFFICIENCY   AND    LOSSES.  491 

of  the  instruments  should  be  equal,  and  the  difference 
between  the  reduced  readings  at  any  other  load  gives 
the  corresponding  drop  of  pressure  in  volts.  The  reduced 
readings  are  gained  by  multiplying  the  readings  of  the 
two  voltmeters  by  their  respective  constants  and  divid- 
ing the  reading  of  the  voltmeter  in  the  high  pressure 
circuit  by  the  ratio  of  transformation  of  the  transformer. 
The  drop  of  pressure,  measured  in  this  way,  includes  the 
CR  drop  in  the  windings  and  the  drop  due  to  magnetic 
leakage,  which  increases  with  the  load.  The  magnetic 
leakage  drop  may  be  determined  by  subtracting  from 
the  total  drop,  the  value  of  the  CR  drop  which  is  calcu- 
lated from  measured  resistances  and  currents. 

A  much  more  accurate  regulation  test  may  be  made 
by  using  two  transformers  of  equal  transformation  ratios 
and  one  voltmeter.  The  primaries  are  separately  con- 
nected to  the  supply  mains,  and  the  secondary  circuits 
are  connected  together  on  one  side.  A  high  resistance 
or  electrostatic  voltmeter  is  connected  between  the  other 
legs  of  the  secondary  circuits.  The  reading  of  this  volt- 
meter at  any  load  on  one  transformer,  when  the  other 
is  unloaded,  gives  the  total  drop  of  pressure  caused  by 
loading  the  former. 

Regulation  tests  are  usually  made  with  non-reactive 
loads,  since  good  regulation  is  a  matter  of  necessity  in 
incandescent  lighting  circuits,  which  are  practically  non- 
reactive.  The  regulation  of  a  transformer  is  changed  for 
the  worse  by  introducing  inductance  into  the  secondary 
circuit,  and  for  the  better  by  introducing  capacity  into 
the  secondary,  as  has  already  been  proven  (Sects.  117  and 
1 1 8).  Regulation  tests  on  a  reactive  load  are  of  little 


492  ALTERNATING   CURRENTS. 

service  except  from  a  comparative  standpoint,  or  to  de- 
termine whether  a  transformer  will  give'  a  satisfactory 
service  to  both  incandescent  lamps  and  alternating-cur- 
rent motors  or  arc  lamps.  The  combined  service  is 
seldom  satisfactory  on  account  of  poor  regulation  which 
injures  the  incandescent  lighting,  though  the  defective 
regulation  is  not  a  matter  of  much  importance  to  the 
power  or  arc-light  service. 

129.  The  All-Day  Efficiency  of  Transformers  and  the 
Effect  of  Magnetic  Reluctance. — The  working  efficiency 
of  a  transformer  is  by  no  means  equal  to  its  full-load 
efficiency.  In  the  case  of  dynamos  placed  in  a  central 
station  it  is  usual  to  divide  the  generating  units  so  that 
the  plant  operating  at  any  part  of  the  day  will  be  fairly 
loaded.  In  the  same  way  the  capacity  of  stationary 
motors,  used  for  distributing  electric  power,  may  be 
chosen  so  that  they  seldom  operate  at  very  small  partial 
loads.  The  way  that  transformers  are  usually  operated, 
however,  makes  it  quite  difficult  to  keep  a  uniform  load 
on  them,  and  in  fact,  for  a  considerable  portion  of  the 
day  they  are  worked  with  the  secondary  circuit  open. 
This  being  the  case,  the  iron  losses  of  transformers  are 
of  much  greater  influence  on  their  all-day  efficiency 
than  are  the  copper  losses,  and  it  is  desirable  to  reduce 
the  iron  losses  to  a  minimum.  For  instance,  suppose  a 
transformer  of  5000  watts  capacity  has  an  iron  loss  of 
2.5  per  cent  or  125  watts,  and  a  copper  loss  at  full  load 
of  2  per  cent  or  100  watts.  Then  the  full-load  effi- 
ciency of  the  transformer  is  95.5  per  cent,  the  half-load 
efficiency  is  94.3  per  cent,  and  the  quarter-load  efficiency 
is  89.5  per  cent,  Supposing  that  the  daily  output  of  the 


EFFICIENCY  AND    LOSSES.  493 

transformer  is  equivalent  to  25,000  watts  for  one  hour 
(25,000  watt-hours),  which  is  equal  to  full-load  operation 
for  five  hours  and  open-circuit  operation  for  the  remain- 
ing 19  hours,  then  the  losses  are  equivalent  in  the  iron 
core  to  3000  watts  for  one  hour,  and  in  the  copper  to 
500  watts  for  one  hour,  or  a  total  of  3500  watts  for  one 
hour.  The  all-day  efficiency  is  then  87.7  per  cent.  To 
increase  this  all-day  efficiency,  it  is  evidently  necessary 
to  decrease  the  iron  losses.  To  do  this  with  a  fixed  fre- 
quency, requires  a  decrease  in  the  amount  of  iron  in  the 
magnetic  circuit  or  a  decrease  of  the  maximum  mag- 
netic density.  Either  process  calls  for  an  increase  in 
the  windings  and  consequently  in  the  copper  losses. 
Suppose  that  decreasing  the  core  losses  to  i \  per  cent 
makes  it  necessary  to  increase  the  copper  losses  to  3 
per  cent;  then,  other  things  being  equal,  the  efficiencies 
become,  full  load,  95.5  per  cent;  half  load,  95.7  per 
cent;  quarter  load,  93.7  per  cent;  and  the  all-day 
efficiency,  on  the  assumption  made  above,  is  90.7  per 
cent.  There  is  a  saving  by  the  latter  construction  of 
950  watt-hours  in  twenty-four  hours,  and  in  one  year 
of  365  days  the  saving  is  nearly  350  kilowatt-hours.. 
If  one  kilowatt-hour  is  worth  10  cents  to  the  central 
station,  the  difference  in  the  amount  of  power  wasted 
each  year  by  the  two  transformers  has  a  value  of 
more  than  thirty-five  dollars,  which  is  several  times 
the  difference  in  the  original  cost  of  the  two  trans- 
formers. If  the  average  load  of  the  transformer  were 
less  than  that  assumed,  which  is  frequently  the  case  in 
practice,  the  iron  losses  would  have  a  still  greater  influ- 
ence on  the  all-day  efficiency.  A  still  greater  decrease 


494         ALTERNATING  CURRENTS. 

in  the  iron  loss  with  its  attendant  increase  of  copper 
loss  would  evidently  raise  the  all-day  efficiency  to  a 
higher  point.  Here,  however,  is  met  the  question  of 
regulation,  which  will  not  satisfactorily  admit  of  too 
great  a  copper  loss  at  full  load  on  account  of  the 
attendant  drop  of  secondary  pressure,  but  this  difficulty 
may  be  met  by  increasing  the  cross-section  of  the  cop- 
per. This  alternative  causes  an  increase  in  the  cost  of 
the  transformer,  but  a  transformer  with  small  losses 
and  good  regulation  is  worth  more  to  the  central  station 
than  one  with  large  losses  or  poor  regulation.  The  all- 
day  efficiency,  upon  "the  assumed  basis,  of  the  17,500 
watt  transformer  previously  referred  to  (page  475)  is 
93.8  per  cent,  and  of  the  6500  watt  transformer  (page 
479)  is  91.3  per  cent.  The  advantage  of  decreasing 
the  iron  losses,  which  is  thus  shown,  led  Swinburne  to 
advocate  and  build  transformers  with  a  cylindrical  iron 
core  under  the  windings,  but  without  closed  iron 
magnetic  circuits.*  Decreasing  the  amount  of  iron  in 
the  magnetic  circuit  decreases  the  core  losses  but 
at  the  same  time  increases  the  reluctance  and  there- 
fore increases  the  magnetizing  current.  This  is  a  de- 
cided disadvantage  if  carried  to  an  excess.  While  the 
true  magnetizing  current  is  wattless,  and  therefore 
requires  the  expenditure  of  no  power  in  the  trans- 
former, yet  it  does  result  in  a  continuous  C^R  loss 
in  the  conductors  leading  to  the  transformer  and  in 
the  primary  winding  of  the  transformer  itself.  It  also 

*  Swinburne,  The  Design  of  Transformers,  Proc.  British  Assoc.  for  the 
Advancement  of  Science,  1888,  and  London  Electrician,  Vol.  23,  p.  492; 
Transformer  Distribution,  Jour.  Inst.  E,  £.,  Vol.  20,  p.  163. 


EFFICIENCY   AND   LOSSES. 


495 


causes  an  extra  demand  for  current  from  the  dynamo 
supplying  the  circuit,  so  that  extra  generators  must  be 
operated  at  periods  of  light  load  in  order  to  supply 
the  wattless  current.  In  other  words,  the  power  factor 
of  the  system  as  a  whole  is  decreased,  with  an  accom- 
panying loss  of  Plant  Efficiency.  Finally,  a  large  mag^ 
netic  reluctance  causes  a  considerable  magnetic  leakage 
and  consequent  increase  in  the  secondary  drop  of  a 
transformer,  and  therefore  interferes  with  its  regulation. 

The  last  two  points  may  become  very  serious  if  the 
reluctance  of  the  magnetic  circuit  of  transformers  is 
made  excessive ;  consequently  Swin- 
burne decreased  the  reluctance  in  his 
transformers  by  making  the  core  of 
iron  wire,  which  was  bent  out  into  a 
spherical  head  (Fig.  228).  From  their 
prickly  appearance  these  transformers 
have  been  called  Hedgehogs.  Figure 
229  shows  the  various  results  of  a  test 
made  by  Fleming  upon  a  3000  watt 
hedgehog  transformer  with  a  ratio  of 
transformation  of  24:1,  and  Fig.  230 
shows  the  power  factor,  at  various 
loads,  of  a  hedgehog  transformer  com- 
pared with  two  closed  iron-circuit  transformers.  It 
has  been  claimed  that  the  transformer  tested  by  Flem- 
ing was  defective  and  the  results  gained  by  Bedell,  in 
testing  a  similar  machine,  were  much  better.* 

Without  entering  a  controversy  regarding  the  exact 


Fig-.  228 


*  Trans.  Amer.  Inst.  E.  E.,  Vol.  10,  p.  497;  Elect.   World,  Vol.  22, 
P-  357- 


496 


ALTERNATING    CURRENTS. 


point  at  which  a  high  reluctance  in  the  magnetic  cir- 
cuit of  a  transformer  causes  more  disadvantage  than  is 


101.9 


95 


.1000  1200  1100  1600  1800  2000  2200  2400  2600 

Fig.  229  a 


3000 


0       200      400      600     800     1000    1200    1400    1600     1800    2000    2200    2400    2000    2880     3000 
OUTPUT  IN  SECONDARY  WATTS 

Fig.  229  b 


DROP 


300         COO        900       1200       1500       1800       2100      2400      2700       3000 
SECONDARY  WATTS 
Fig".  229  C 

counterbalanced  by  the  decreased  iron  losses  brought 
about  by  decreasing  the  amount  of  iron,  the  examples 
may  serve  to  show  the  necessity  of  carefully  designing 


EFFICIENCY   AND    LOSSES. 


497 


the  magnetic  circuit  to  fit   the  conditions.       Where  a 

O 

fairly  large  proportion  of  the  full  load  is  carried  by 
the  transformer  a  great  portion  of  the  time,  as  may 
be  the  case  in  many  of  the  proposed  power  transmis- 
sions, the  reluctance  of  the  magnetic  circuit  of  the 
transformer  should  be  made  very  small,  so  that  the 
copper  losses  may  be  small.  On  the  other  hand,  where 
the  average  load  is  very  small,  as  in  the  ordinary 


.1  -2  .3  .4  .5  .6  .1  .8  .9  '.10 

FRACTIONS  OF  FULL   LOAD 

Fig.  23O 

arrangements  of  transformers  for  lighting  service,  the 
CZR  loss  is  of  less  moment  than  the  iron  losses,  and 
every  effort  must  be  made  to  decrease  the  amount  of  iron, 
and  hence  the  iron  losses,  as  far  as  the  requirements 
of  economy  of  construction,  plant  efficiency,  and  regula- 
tion will  permit.  Transformers  might  even  be  built 
without  iron  in  the  core,  were  it  not  for  the  immoder- 
ate cost  caused  by  the  large  amount  of  copper  which 
would  be  required  in  their  construction  to  gain  a  rea- 
sonably good  degree  of  regulation. 


2K 


498  ALTERNATING   CURRENTS. 

The  unsatisfactory  features  of  transformers  having 
cores  of  high  magnetic  reluctance  may  be  largely  neu- 
tralized by  the  use  of  condensers  (Sect.  35).  These 
may  be  put  in  parallel  with  each  transformer  of  such 
a  capacity  as  to  practically  supply  the  necessary  watt- 
less magnetizing  current,  or  a  group  of  condensers 
may  be  connected  in  parallel  to  each  circuit  of  suffi- 
cient capacity  to  supply  the  wattless  current  for  that 
circuit.  In  either  case,  the  difficulties  due  to  regula- 
tion and  plant  efficiency  are  obviated.  Tests  of  con- 
densers in  parallel  with  transformers  have  shown  that 
their  use  is  entirely  practical,  provided  that  a  cheap  and 
durable  dielectric  can  be  found.* 

130.  Curves  of  Efficiency. — The  curve  showing  the 
efficiencies  of  a  good  transformer  at  various  loads  is  a 
very  flat  one.  In  Fig.  231  are  given  the  curves  of 
a  4000  watt  Thomson-Houston  transformer  tested  by 
Fleming,  the  17,500  watt  transformer  tested  by  Ryan, 
which  already  has  been  referred  to  (p.  4/5),  and  the 
3000  watt  hedgehog  transformer  tested  by  Bedell, 
which  was  referred  to  directly  above  (p.  495).  These 
curves  are  flatter  than  the  similar  curves  for  dynamos 
(Vol.  I.,  p.  270).  To  get  the  best  all-day  efficiency, 
it  is  evident  that  every  effort  should  be  bent  to  mak- 
ing the  knee  of  the  curve  at  the  smallest  possible  load, 

*  Swinburne,  The  Probable  Use  of  Condensers  in  Electric  Lighting, 
London  Electrician,  Vol.  28,  p.  227 ;  Le  Blanc,  Application  of  Alternating 
Currents  to  the  Transmission  of  Power,  Soc.  Inter,  des  Electriciens,  1891; 
and  London  Electrician,  Vol.  27,  p.  383;  Swinburne,  Transformer  Distri- 
bution, Jour.  Inst.  E.  E.,  Vol.  20,  p.  191,  and  London  Electrician,  Vol.  26, 
p.  548  :  Bedell  and  others,  Hedgehog  Transformers  and  Condensers,  Pt.  3, 
Trans.  Amer.  Inst.  E.  £.,  Vol.  10,  p.  513. 


EFFICIENCY   AND   LOSSES. 


499 


II 
o  if 
If 


§ 

•AO.N3IOIJJ3 


500  ALTERNATING   CURRENTS. 

without  at  the  same  time  in  creasing  the  exciting  current 
too  greatly.  The  maximum  all-day  efficiency  for  a  given 
transformer  comes  at  such  a  distribution  of  the  load  that 
the  watt-hours  represented  by  the  copper  and  iron  losses 
are  equal.  The  maximum  operating  efficiency  occurs  at 
the  load  which  causes  copper  and  iron  losses  to  be  equal. 
The  full-load  efficiency  of  average  commercial  trans- 
formers of  different  sizes  is  represented  by  the  curve 
of  Fig.  232.  These  efficiencies  may  be  improved  upon, 
but  they  represent  average  practice. 

131.  Weight  Efficiency.  —  The  total  weight  of  trans- 
formers is  exceedingly  variable,  as  it  depends  not  only 
upon  the  design  of  the  machine,  but  also  upon  the  con- 
taining case.     The  weights  of  copper  and  iron  depend 
entirely    upon    the    purpose   of   the    designer,  and   the 
limits  to  which  he  has  carried  a  desire  to  gain  a  high 
all-day  efficiency.     The  total  weights  of  transformers  de- 
signed for  frequencies  not  less  than  100,  nor  greater  than 
135,  will  ordinarily  vary  between  75  and  100  pounds  per 
kilowatt  for  transformers  near  one  kilowatt  in  capacity, 
and  from  40  to  60  pounds  per   kilowatt    for   transfor- 
mers of  10  K.W.  capacity.     Figure  233  gives  the  total 
weights  of  the  standard  transformers  of  two  well-known 
manufacturers,  which  are  designed  for  a  frequency  of 
135  and  Fig.  234  for  frequencies  of  30  to  60. 

132.  Separation  of  Core  Losses. — The  hysteresis  loss 
in  a  transformer  may  be  considered  constant  for  a  given 
frequency  and  pressure,  as  indicated  by  Ewing's  neat 
experiment  (Sect.  127),  but  the  foucault  current  loss  will 
become  less  after  the  transformer  has  been  run  under 
load  and  thus  become  heated  up ;  for,  as  the  core  rises 


EFFICIENCY   AND    LOSSES. 


501 


g 
oven  -nnj  j.v  - 


502  ALTERNATING   CURRENTS. 

in  temperature,  the  resistance  of  the  iron  increases. 
To  separate  the  hysteresis  from  the  foucault  current 
loss,  the  iron  losses  may  be  measured  when  the  core 
is  cold  (Sect.  125  (8 a)),  and  then  when  it  has  become  hot 
by  being  run  under  load.  Let  Wc  be  the  first  reading, 
and  Wh  the  second  ;  also,  let  f  be  the  difference  of 
temperature  of  the  core  at  the  two  readings.  Then 

Wc  =  H  +  F,    and     Wh  =  H  -j-  Fh, 

where  H  is  the  hysteresis  loss,  and  Fe  and  Fh  the  foucault 
current  losses,  cold  and  hot.  The  foucault  current  loss  is 

F=  CE  =  — ,  where  C  and  E  are  a  current  and  pressure 
R 

equivalent  to  the  foucault  currents  and  the  pressures 
acting  upon  them,  and  R  is  a  resistance  equivalent  to 
that  of  the  foucault  current  circuits  combined.  E  is 
constant  during  the  test,  as  it  depends  only  upon  the 
magnetic  density,  the  primary  pressure,  the  frequency, 
and  the  dimensions  of  the  core  plates.  Then 

E2  E* 

Fc  =  — ,   and    Fh  —  —  - — -  — —  > 

where  Rc  is  the  resistance  at  the  temperature  of  the  cold 
measurement,  and  a  is  the  temperature  coefficient  of 
the  iron  comprising  the  core  discs.  From  the  value  of 
Wc  and  Wv  we  have  Wc  -  Wh  =  Fc  —  Fh,  and  substitut- 
ing the  values  of  Fc  and  Fny 


w       w_ 

h~  Rc 


& 

and  as  ~  =  Fc, 


EFFICIENCY   AND   LOSSES. 


503 


As  all  the  quantities  in  the  right-hand  side  of  the  equa- 
tion are  determined  by  measurement,  the  foucault  cur- 
rent loss  may  be  thus  separated  from  the  hysteresis  loss. 


-*  b3d  -sai 


The  coefficient  a  may  be  readily  obtained  by  measuring 
the  resistance  between  any  two  points  on  a  core  disc 
when  the  core  is  at  two  different  temperatures,  when  the 
quantity  desired  will  be  the  per  cent  change  in  resist- 


504  ALTERNATING   CURRENTS. 

ance  per  degree  change  in  temperature.  In  order  that 
the  results  of  these  measurements  may  be  reasonably 
reliable,  the  pressure  applied  to  the  transformer  during 
the  tests  must  be  exceedingly  constant.  The  foucault 
current  loss  in  commercial  transformers  is  but  a 
small  proportion  of  the  total  core  loss.  The  hystere- 
sis and  foucault  current  losses  may  also  be  separated  by 
measuring  the  core  losses  at  two  frequencies  for  the 
same  pressure,  then,  by  subtraction  and  an  elimination 
similar  to  that  given  in  Vol.  I.,  p.  254,  the  value  of  the 
foucault  current  loss  may  be  deduced. 

133.  Tests  of  American  Transformers.  —  The  most 
complete  public  test  of  recent  American  transformers 
is  one  made  by  Mr.  A.  H.  Ford,  in  the  laboratories  of 
the  University  of  Wisconsin,  during  the  year  1895.* 

The  list  included  transformers  of  the  following  types  : 
Bullard,  Diamond,  Fort  Wayne,  General  Electric,  Horn- 
berger,  National,  Packard,  Phoenix,  Standard,  Stanley, 
Wagner,  and  Westinghouse.  The  total  number  of 
transformers  tested  was  over  twenty. 

In  making  the  tests,  Weston,  Hoyt,  Queen,  and 
Cardew  instruments  were  used,  and  their  accuracy  was 
frequently  checked  by  comparison  with  Kelvin  bal- 
ances. Great  care  was  exercised  to  eliminate  errors. 
The  tests  were  nfade  at  frequencies  of  60  and  125  ;  the 
pressure  wave  being  rather  peaked,  especially  at  the 
frequency  of  60. 

The  method  used  for  finding  the  efficiency  was  that 
given  in  Section  125,  No.  8  a,  and  the  results  were 

*  Complete  Tests  of  Modern  American  Transformers,  Bull.  Univ.  of 
Wisconsin,  Vol.  i,  No.  n. 


EFFICIENCY   AND    LOSSES. 


505 


checked  by  methods  8  b  and  7.  Method  8  a  was  found 
to  be  the  most  accurate,  but  by  using  other  methods  as 
checks  serious  errors  could  be  detected  and  the  test 
repeated.  As  the  method  used  measures  the  losses  di- 
rectly, small  errors  of  observation  cause  an  inappreciable 
error  in  the  result.  The  all-day  efficiencies  were  calcu- 
lated on  the  assumption  that  during  each  24  hours  the 
transformer  runs  5  hours  at  full  load,  and  19  hours  at  nc 
load.  This  assumption  gives  about  the  proper  all-day 
efficiency  to  represent  favorable  conditions  of  present 
practice.  The  following  table  gives  a  synopsis  of  the 
results  obtained  from  twelve  of  the  transformers  : 

TABLE  I. 


No. 

Capacity 
in 

Iron  Loss. 

Copper 
Loss. 

Maximum 
Efficiency. 

All-Day 
Efficiency. 

Watts. 

/=  125. 

/=6o. 

7=125- 

/=6o. 

/=  125. 

/=6o. 

, 

1250 

37-o 

48.5 

29.7 

95-° 

94.0 

85.I 

83.0 

2 

1500 

5°-5 

70.6 

45-2 

94.6 

94.0 

84.8 

80.6 

3 

1500 

32.2 

46.5 

38.1 

95-7 

94-5 

89.4 

85-0 

4 

1500 

57-o 

82.0 

35-3 

93-7 

92.0 

83.0 

77.8 

5 

1500 

45-7 

60.  1 

36-2 

94-8 

94.0 

85.5 

83.4 

6 

1500 

126.0 

206.0 

14.8 

9i-5 

87.4 

70.8 

60.0 

7 

45° 

23-4 

38.6 

6.2 

91.0 

76-5 

8 

1800 

53-5 

108.7 

66.6 

94.0 

91.7 

84.7 

75-5 

9 

2OOO 

42.4 

56.3 

54-8 

95-2 

94-5 

88.6 

86.5 

10 

I5OO 

97-5 

125.0 

38.5 

91.7 

90.1 

76.5 

70.2 

ii 

I5OO 

57-5 

76.0 

30.9 

94-5 

93-4 

83.0 

79.2 

12 

I5OO 

43-2 

55-5 

28.5 

95-3 

94.6 

86.5 

83-7 

Some  of  the  transformers  giving  high  maximum  effi- 
ciencies do  not  give  the  highest  all-day  efficiencies,  and 
in  such  cases  the  figures  would  tend  to  indicate  that  an 


506 


ALTERNATING   CURRENTS. 


increase  in  the  number  of  turns  and  a  decrease  in  the 
iron  and  magnetic  density  would  be  of  advantage. 

The  iron  losses  are  so  variable  that  a  table  was  made 
up  of  the  losses  per  cubic  centimeter,  from  which 
rather  better  comparisons  can  be  made.  As  the  mag- 
netic densities  in  the  transformers  are  very  different, 
this  does  not  directly  give  the  relative  qualities  of  the 
iron,  hence  another  column  is  added  of  the  hysteresis 
constants.*  These  constants  are  calculated  from  the  for- 
mula of  Steinmetz,  U  —  vVB™  (see  Vol.  I.,  p.  74),  where 
U  is  energy  in  watts  lost  per  cubic  inch  or  per  cubic 
centimeter  of  iron,  V  is  the  frequency,  B  the  maximum 
induction  (per  square  centimeter),  and  v  is  a  constant  of 
hysteresis  which  depends  upon  the  quality  of  the  iron. 
Table  II.  presents  the  reduced  results. 

TABLE   II. 


No. 

Loss  per  cu.  in. 

Loss 
per  cu.  cm. 

Bm 

Constant  of  Hysteresis. 

/=i25. 

7=6o. 

/=i25. 

7=6o. 

f=  125- 

7=6o. 

7=  "5. 

7=60. 

I 

•13 

.18 

.008 

.Oil 

2050 

4280 

3.2  x  io~10 

2.9  x  io~10 

2 

.21 

.28 

.013 

.017 

2600 

5400 

3-52X  io~10 

3.04  x  io~10 

3 

4 

.24 

•34 

.014 

.O2I 

5 

.24 

•32 

•015 

.O2O 

3640 

7500 

2.39X  io~10 

2.IOX  IO~10 

6 

.68 

I.  10 

.041 

.067 

3750 

7720 

6.26xio-10 

6.57x10-!° 

7 

.27 

•45 

.017 

.027 

3770 

7870 

2-59X  io~10 

2.66  xio-10 

8 

.20 

.40 

.013 

.025 

525° 

10950 

9 

.26 

•34 

.017 

.021 

3540 

737° 

2.88XIO-10 

2.86x10-1° 

10 

.40 

•51 

.024 

.031 

ii 

•30 

•39 

.018 

.024 

4070 

8480 

2.42X  IO-10 

2.o8x  iQ-10 

12 

•23 

•30 

.014 

.018 

3210 

6650 

2-72X  IQ-10 

2.25X  I0~10 

In  calculating  the  hysteresis  constants,  the  core  losses 
were  considered  to  be  entirely  due  to  hysteresis,  which 


EFFICIENCY    AND    LOSSES. 


507 


does  not  introduce  a  large  error,  as  the  foucault  cur- 
rent loss  is  only  a  small  portion  of  the  total  loss.  A 
glance  indicates  the  great  difference  in  the  quality  of 
iron  used  in  the  different  transformers,  and  shows  very 
distinctly  the  necessity  for  testing  each  batch  of  iron 
before  it  is  made  up  into  transformer  cores.  As  the 
iron  losses  are  the  most  important  factor  in  determin- 
ing the  all-day  efficiency,  too  much  stress  cannot  be 
laid  upon  this  point.  The  table  indicates  that  the  prac- 
tice of  making  such  tests  has  not,  by  any  means,  been 
universal. 

Table  III.    gives  the  exciting  currents  and  no-load 
power  factors  of  the  transformers.     The  power  factors 

TABLE   III. 


No. 

Exciting  Current. 

Power  Factor,  No  Load. 

Bm 

/=  125- 

y=6o. 

/=  "5. 

/=6o. 

/=»5. 

/=&>. 

! 

•043 

.066 

84.0 

73-o 

2O5O 

4280 

2 

.076 

.124 

64.6 

56.5 

2600 

5400 

3 

.t>52 

.100 

56.3 

47.6 

4 

.085 

.125 

67.0 

65.6 

5 

•054 

.099 

8l.7 

61.5 

6 

•173 

•475 

85.0 

40.0 

3750 

7720 

7 

•043 

.079 

64.0 

58.0 

377° 

7870 

8 

.076 

.060 

7I.O 

22.0 

5250 

10950 

9 

•055 

.091 

78.5 

63.0 

3540 

737° 

10 

.124 

.190 

78.6 

65.7 

ii 

.072 

•"3 

79.6 

67.2 

4070 

8460 

12 

.077 

.144 

55-6 

38.4 

3210 

6650 

are  higher  at  the  higher  frequency,  due  to  the  fact  that 
a  smaller  magnetizing  current  is  required  on  account  of 


508 


ALTERNATING   CURRENTS. 


the  magnetic  density  being  lower  at  that  frequency. 
The  exciting  current  is  lower  at  the  higher  frequency, 
due  to  the  above  cause  and  to  the  lower  iron  losses. 

Table  IV.  gives  the  results  of  an  independent  test 
of  the  regulation  of  the  transformers.  The  total  drop 
is  the  difference  in  volts  between  the  secondary  press- 
ure at  full  load  and  at  no  load  when  the  primary 


TABLE   IV. 


No. 
I 

Volts 
Total  Drop. 

Volts 
C7?Drop 
in  Secondary. 

Volts 
CR  Drop 
in  Primary. 

Volts 
Leakage  Drop. 

/=  '25- 

/»<x* 

/=  "5- 

/=6o. 

/=i2S. 

/=6o. 

./=  I25- 

/=6o. 

2-3 

3-5 

.80 

1.  00 

10.9 

12.4 

•5 

1-3 

2 

3-5 

2.8 

.85 

.88 

7-9 

7-7 

1.9 

i.i 

3 

4.9 

3-3 

•°5 

I.IO 

12.6 

13.0 

2.6 

•9 

4 

4.6 

3-4 

•35 

!-35 

8.5 

8.5 

2.4 

1.2 

5 

2.9 

2.O 

.04 

.98 

9-3 

8.7 

I.O 

.1 

6 

1.8 

1.2 

.41 

.42 

3-4 

3-9 

I.I 

•4 

7 

82.0 

21.0 

•94 

2.48 

24.2 

28.0 

77-7 

15-7 

8 

5-2 

4-7 

.40 

1.40 

17.0 

17.8 

2.1 

i-5 

9 

5-3 

4.2 

•7i 

*-75 

8,2 

8.4 

2.8 

i-7 

10 

4.0 

3-1 

ii 

2.8 

2.2 

1.04 

1.04 

8.2 

7-9 

I.O 

•4 

12 

3-8 

2.O 

1.36 

1.36 

12.7 

12.5 

1.2 

pressure  is  1000  volts.  In  one  of  the  transformers 
it  is  seen  that  the  total  drop  is  greater  for  a  frequency 
of  60  than  of  125,  while  in  the  others  the  opposite 
is  the  case.  The  leakage  drop  should  be  less  at  the 
lower  frequency,  as  the  total  number  of  leakage  lines 
does  not  change  very  much  with  the  change  in  the 
frequency,  but  the  inductive  pressure  varies  directly 


EFFICIENCY   AND   LOSSES.  509 

with  the  frequency.  This  is  shown  very  plainly  in 
transformer  No.  7,  in  which  the  magnetic  leakage  is 
sufficiently  great  to  cause  a  current  of  almost  constant 
value  to  flow  in  the  secondary  circuit  when  the  fre- 
quency is  125,  but  when  the  frequency  is  60  it  fails  of 
its  purpose  entirely.  The  characteristic  curves  of  this 
transformer  are  given  in  Fig.  204. 

The  leakage  drop  is  somewhat  less  for  the  transform- 
ers which  have  the  apertures  for  windings  with  a  larger 
dimension  in  the  direction  of  the  lines  of  force  in  the 
tongue  of  the  transformer  and  with  the  coils  wound 
one  upon  the  other  (see  Figs.  240  and  241).  A  compari- 
son of  the  CR  drops  shows  that  the  transformers  of 
high  all-day  efficiencies  have  a  comparatively  large  fall 
of  pressure  from  this  cause. 

Table  V.  gives  the  radiating  surface  in  the  trans- 
formers and  the  rise  of  temperature  under  various 
conditions  measured  in  a  special  test.  From  the  table 
it  is  evident  that  there  is  little  uniformity  in  the  watts 
radiated  per  square  inch  of  core  and  coil,  or  case,  per 
degree  rise  of  temperature.  Taking  an  average  of  the 
test,  the  watts  radiated  per  square  inch  of  the  core  and 
coil  surface  with  the  case  on  may  be  about  .2,  and  with 
the  case  off  .3,  when  the  rise  of  temperature  is  about 
40°  C.  This  requires  from  5  to  7  inches  of  core  and 
coil  surface  per  watt  lost  with  the  case  on.  In  an  ex- 
cellent series  of  transformers  of  various  capacities  the 
product  of  rise  of  temperature  and  square  inches  of 
coil  and  core  per  watt  radiated  varies  between  about 
270  and  360,  with  an  average  of  about  310.  This  is 
nearly  50  per  cent  greater  than  the  average  result  of 


510 


ALTERNATING   CURRENTS. 


TABLE  V. 


6 
fe 

Total 
Watts  Lost. 

Watts  Lost 
per  sq.  in. 
of  Case. 

jlJ 

Rise  of 
Temperature 
in  Case. 
Centigrade. 

Frequency,  _/". 

Current  in 
Secondary. 

its? 

I 

SI 

.105 

.121 

20.8 

60 

O 

2-5 

I 

39 

.080 

•093 

17-5 

125 

O 

41.0 

I 

82 

.168 

•195 

50.2 

60 

I3-I 

5i 

I 

65 

•134 

•155 

38.4 

I25 

"•3 

24 

2 

75-3 

.16 

16.2 

60 

0 

} 

2 

53-8 

.11 

13-4 

125 

0 

\    Taken  without 

2 

XI5-3 

.24 

50-4 

60 

17.9 

f    case. 

2 

93-8 

.20 

51.4 

125 

17.6 

J 

3 

49.6 

.082 

20.0 

60 

O 

43 

3 

34-5 

.058 

I7.8 

125 

O 

178 

3 

85.6 

.144 

56-3 

60 

14.2 

— 

3 

59-5 

.100 

36.0 

I25 

12.5 

89 

a 

233 

.225 

.396 

73-4 

60 

0 

66 

a 

143 

.178 

•243 

66.1 

125 

0 

no 

a 

275 

•34° 

.466 

IOO.O 

60 

18.7 

55 

a 

198 

.242 

•334 

70.0 

125 

19-5 

63 

5 

64 

.118 

.166 

21.8 

60 

O 

— 

5 

49 

.090 

.127 

19.1 

125 

O 

— 

5 

93 

.172 

.242 

40.8 

60 

13-7 

— 

5 

78 

.144 

.203 

40.6 

I25 

13.6 

— 

6 

2IO 

.388 

•547 

62.4 

60 

0 

25 

6 

133 

.246 

•346 

52.3 

125 

0 

114 

6 
6 

22| 
246 

.412 
•455 

•580 
.640 

86.8 
72.2 

60 
125 

14.7 

14.6 

20 
23 

7 

41 

•143 

•175 

3M 

60 

0 

12.5 

7 

25 

.091 

.107 

24-3 

125 

0 

14.6 

7 

46 

.162 

.198 

57-4 

125 

8.6 

12 

8 

JI5 

.168 

.266 

43-4 

60 

O 

49 

8 

56 

.079 

.130 

32.1 

125 

O 

33-8 

8 

171 

.250 

•396 

101.8 

60 

17.1 

5-3 

8 

102 

.150 

•234 

76.9 

125 

15-7 

— 

9 

60 

.099 

.150 

25-4 

60 

0 

1.2 

9 

45 

.074 

.112 

21.2 

125 

O 

48 

9 

IO2 

.168 

•255 

67.5 

60 

16.9 

10 

9 

90 

.149 

.225 

51.6 

125 

17.6 

— 

£ 

29 

.052 

.110 

20.1 

60 

0 

37-6 

b 

26 

.047 

.098 

15.2 

125 

0 

ii.  8 

b 

57 

.102 

.214 

47.8 

60 

9-4 

15-5 

b 

47 

.085 

.190 

30.8 

125 

8-3 

— 

ii 
ii 
ii 
ii 

76 

57-5 
103.8 
88.4 

.274 
.208 
•372 
.320 

27.0 
18.9 
52.2 
50-4 

60 
125 
60 
125 

0 
0 

14.3 
15-1 

!    Taken  without 
case.     Areas 
core  and  coil 
277  sq.  in.,  and 
of  case  514. 

12 

55-5 

.086 

•145 

31-6 

60 

O 

60 

12 

43-2 

.067 

•IT3 

2O.5 

125 

0 

67 

12 

80.8 

.125 

.211 

SI.8 

60 

14.1 

— 

12 

78.8 

.122 

.206 

20-5 

125 

14.2 

~ 

EFFICIENCY   AND    LOSSES.  511 

Mr.  Ford's  tests.  These  figures  are  very  rough  approx- 
imations, to  be  used  merely  as  a  guide  in  design  ;  but 
fairly  exact  data  for  any  particular  type  of  transformer 
may  be  determined  from  measurements  on  two  or  three 
sizes  of  the  type. 

The  transformer  case  increases  the  temperature  on  an 
average  of  about  43  per  cent ;  but  this  effect  is  found 
to  be  quite  variable.  The  form  of  the  case  does  not 
seem  to  enter  into  the  result,  though  when  it  closely 
encases  the  core  and  coils  the  heating  effect  seems  to 
be  slightly  less. 

The  table  shows  the  results  of  four  heating  tests  on 
each  transformer ;  a  no-load  and  a  full-load  test  at  each 
of  the  two  frequencies  used ;  the  temperatures  being 
determined  from  measurements  of  the  resistance  of  the 
primary  coils,  ample  time  being  allowed  in  each  test  for 
the  transformer  to  come  to  its  maximum  temperature, 
and  check  measurements  being  made  by  thermometer. 

The  following  table,  from  a  test  by  Fleming,*  made  in 
1892,  gives  the  principal  data  concerning  a  number  of 
English  transformers  and  two  of  American  make. 

A  is  the  capacity  in  watts,  B  the  exciting  current  in 
amperes,  with  a  primary  pressure  of  2400  volts  and  fre- 
quency of  83,  C  the  iron  loss  in  watts,  D  the  power 
factor  at  no  load,  E  the  total  drop  in  volts,  F  the  cop- 
per drop  in  volts,  and  G  the  leakage  drop  in  volts.  The 
low  power  factor  and  high  exciting  current  of  the  Swin- 
burne hedgehog  transformer  is  especially  noteworthy. 
This  is  caused  by  the  high  reluctance  of  the  open  mag- 
netic circuit. 

*  Jour.  Inst.  E,  E.,  Vol.  21,  p.  594;   Electrical  World,  Vol.  21,  p.  15. 


512 


ALTERNATING  CURRENTS. 


Name. 

^ 

B 

c 

D 

E 

F 

^ 

Kerranti 

iSvc 

18 

288 

66 

« 

37^0 

•7A 

C.4O 

68 

i  6 

I  Q 

« 

JO'-' 
7  coo 

2C 

yt** 

A  A  A 

7A 

«« 

11250 

•»3 

•  ^4- 

578 

•/'t 
.70 

it 

I5OOO 

.  ^7 

IOIQ 

7C. 

« 

37C.O 

.11 

211. 

88 

2  4. 

2  I 

-? 

« 

7^00 

O7 

iiS 

77 

« 

Fir*" 

I  I2CO 

O7 

1^.8 

81 

3       A 

2  7 

7 

« 

I  COOO 

j  i 

228 

gc 

•4 
2   I 

*•  / 

i  6 

r 

« 

II2O5 

IO 

228 

•°J 
Q2 

2  2 

1.8 

O 

•  4. 

Swinburne  Hedgehog 
<« 

3000 
6OOO 

•74 

I   IQ 

112 

K6 

.06 

oc 

3-2 

2.2 

1.0 

Westinghouse     .   .   . 

Mordey-Brush    .  .   . 
« 

Thomson-Houston   . 
Kapp 

6500    . 
6000 

750 
4500 
4.OOO 

•05 
.08 

•°3 
.08 

.14 

95 
140 
61 
108 

IC2 

»V3 

•79 
•77 
.81 

•54 
.61 

2.4 

1.8 
3-3 

I.Q 

I.4 

1.8 

2-5 

i  8 

I.C 

.8 

DESIGN   OF   TRANSFORMERS.  513 


CHAPTER    XII. 

DESIGN    OF    TRANSFORMERS. 


134.    Effect  of  Frequency.  —  The  formulas  giving  the 
pressures  developed  in  the  coils  of  a  transformer 

,    _..     n"  „,     /c,  x 

and  £"  =  —  .£'    (Sect,  in) 


g 

show  that  if  the  value  of  the  magnetization  in  the  core 
of  a  transformer  is  fixed,  then  the  number  of  turns  in 
the  coils  must  vary  inversely  with  the  frequency.  To 
construct  transformers  for  all  frequencies  with  a  fixed 
value  of  the  total  magnetization  is  not  an  economical 
plan,  however,  since  the  core  loss  per  cubic  centimeter 
of  iron  depends  upon  the  frequency.  If  a  certain  core 
loss  is  determined  upon  as  being  that  which  will  give 
the  most  satisfactory  general  results  in  the  case  of  a 
transformer  of  given  size,  the  magnetic  density  in  the 
core  must  be  made  to  vary  inversely  with  the  frequency. 
This  may  be  seen  from  the  fact  that  the  foucault  cur- 
rent loss  varies  with  the  square  of  the  frequency,  and 
the  hysteresis  loss  directly  with  the  frequency.  Since 
the  former  is  between  one-tenth  and  one-fourth  of  the 
latter,  this  makes  the  total  core  losses  vary  somewhere 
between  the  first  and  second  powers  of  the  frequency,  if 
the  magnetic  density  is  constant.  The  foucault  current 

2L 


514         ALTERNATING  CURRENTS. 

loss  also  varies  with  the  square  of  the  magnetic  den- 
sity, and  hysteresis  varies  with  some  degree  of  approxi- 
mation within  the  limits  of  transformer  practice,  as  the 
1.6  power  of  the  induction.  Consequently  the  total 
losses  vary  as  some  power  of  the  magnetic  density 
between  the  i.6th  and  the  second.  It  is  thus  shown 
that  the  core  losses  will  not  be  greatly  changed  if  the 
magnetic  density,  within  reasonable  limits,  varies  in- 
versely with  the  frequency.  Now  it  is  seen  from 
the  formula  that,  if  the  turns  and  pressures  remain 
unchanged,  when  the  frequency  changes,  the  induction 
will  change  inversely ;  hence  a  well-built  transformer, 
should  give  nearly  the  same  efficiency  for  all  frequen- 
cies reasonably  near  that  for  which  it  was  designed, 
and  the  number  of  turns  in  the  coils  may  be  the  same 
in  a  series  of  transformers  of  equal  capacities  designed 
for  different  frequencies.  In  support  of  this  stands  the 
fact  that  transformers  properly  built  for  125  to  135 
periods  a  second  are  capable  of  giving  excellent  results 
at  a  frequency  of  60,  as  is  indicated,  for  instance,  in  a 
number  of  the  transformers  as  given  in  Table  I.  of  the 
preceding  section.  Changing  the  frequency  alters  the 
wattless  component  of  the  exciting  current  in  a  propor- 
tion which  depends  upon  the  saturation  of  the  core, 
and  poorly  designed  transformers  or  those  built  of  poor 
material  may  give  unsatisfactory  service  and  over-heat 
on  lower  frequencies  than  their  normal,  on  account  of 
an  excessive  magnetic  density  and  an  excessive  exciting 
current. 

Since  the  rates  of  variation  of  the  hysteresis  and  fou- 
cault  current  losses  with  the  frequency  and  with   the 


DESIGN   OF    TRANSFORMERS. 


515 


reciprocal  of  the  magnetic  density  are  not  exactly  the 
same,  though  they  are  quite  approximately  equal,  it  is  to 
be  expected  that  the  efficiency  of  a  transformer  will 
vary  to  some  degree  with  the  frequency,  and  that  some 
particular  frequency  will  give  the  best  efficiency.,  The 
latter  has  been  found  to  be  the  case  by  Mr.  Mordey.* 


75 


125 


100 


HOURS.      ,1234  567 

Fig.  235 

Figure  235  shows  the  curves  of  rise  in  temperature  of 
a  Mordey  transformer  tested  at  three  different  frequen- 
cies, showing  that  the  best  result  is  given  for  an  interme- 
diate value.  The  exact  frequency  at  which  a  transformer 
will  give  the  highest  efficiency  must  depend  upon  the 
quality  of  the  iron  and  the  effectiveness  of  the  lamina- 

*  Alternate  Current  Working,  Jour.  Inst.  E.  £.,  Vol.  1 8,  p.  608;  Dis- 
cussion of  papers  by  Snell  and  Forbes,  Jour.  Inst.  E.  £.,  Vol.  22,  pp. 
280  and  484. 


5l6  ALTERNATING   CURRENTS. 

tion,  since  these  determine  the  rate  of  variation  of  the 
losses  with  the  magnetic  density  and  the  frequency. 
The  better  the  magnetic  quality  of  the  iron  and  the 
more  thorough  its  lamination,  the  less  difference  is  likely 
to  occur  in  the  efficiencies  at  various  frequencies.  A 
transformer  made  of  poorly  laminated  iron  is  likely  to 
give  its  best  efficiency  at  a  moderate  frequency,  while 
one  made  of  well  laminated  iron  is  likely  to  give  its  best 
efficiency  at  a  higher  frequency.  In  first-class  modern 
commercial  transformers  the  effect  of  foucault  currents 
on  the  efficiency  is  quite  small,  so  that  there  is  no  real 
maximum  point  in  the  relation  between  efficiency  and 
frequency,  but  the  efficiency  increases  slightly  but  con- 
tinuously as  the  frequency  increases  within  commercial 
limits.  Roughly,  the  core  losses  may  be  said  to  vary 
inversely  as  the  square  root  of  the  frequency. 

It  is  possible  to  build  different  transformers  of  the 
same  number  of  turns  for  different  frequencies  with 
the  cross-section  of  the  core  inversely  proportional  to 
the  frequency.  The  magnetic  density  would  then  be 
the  same  in  all  the  transformers.  This  is  unsatisfac- 
tory, however,  since  it  makes  low  frequency  transfor- 
mers large  and  expensive :  and  since  the  weight  of 
iron  in  a  core  must  vary  more  rapidly  than  the  cross- 
section,  this  method  of  construction  also  causes  com- 
paratively large  core  losses  in  low  frequency  trans- 
formers and  gives  them  a  comparatively  low  efficiency. 

The  conclusion  is  therefore  derived  that  the  core  and 
windings  of  a  well  designed  and  well  constructed  trans- 
former will  be  almost  equally  satisfactory  in  performance 
on  any  frequency  within  the  present  commercial  limits. 


DESIGN   OF   TRANSFORMERS.  517 

This  conclusion  is  shown  by  experience  to  be  approxi- 
mately true  for  frequencies  within  the  limits  of  50  and 
150.  The  table  of  satisfactory  magnetic  densities  for 
various  frequencies,  already  given  (Sect.  97),  shows 
that  foreign  manufacturers  apparently  prefer  a  mean 
path,  and  change  both  the  cross-section  of  the  core  and 
the  induction  when  building  transformers  for  different 
frequencies.  American  manufacturers  have  heretofore 
ordinarily  built  transformers  of  one  standard  type,  which 
are  intended  to  be  used  on  all  usual  commercial  fre- 
quencies, but  some  are  now  building  two  types,  one 
designed  specially  for  frequencies  above  100,  and  the 
other  for  frequencies  between  50  and  100.  The  latter 
have  a  somewhat  larger  core.  The  outputs  of  trans- 
formers of  a  fixed  size  are  almost  in  proportion  to  the 
square  root  of  the  frequency,*  other  things  being  equal. 
As  the  frequency  decreases,  the  magnetic  density  may 
be  allowed  to  increase  to  as  much  as  12,000  lines  of 
force  per  square  centimeter  in  large  transformers  of  low 
frequency.  This  density  may  be  satisfactorily  used  in 
transformers  of  30  frequency,  and  those  designed  for 
lower  frequencies  must  increase  in  weight  in  inverse 
proportion  to  the  frequency. 

135.  Effect  of  the  Form  of  the  Impressed  Pressure 
Curve.  —  Very  few  experimental  data  are  at  hand  which 
show  the  effect  of  the  form  of  the  pressure  curve, 
though  a  good  deal  has  been  written  upon  the  subject. 
No  matter  what  the  primary  pressure  wave  may  be  like, 
the  secondary  pressure  wave  must  have  practically  the 

*  Steinmetz,  Effect  of  Frequency  on  the  Output  of  Transformers,  Elect. 
Engineer,  Vol.  15,  p.  260;  Trans.  Amer.  Imt.  E.E.,  Vol.  n,  p.  38. 


5i8 


ALTERNATING    CURRENTS. 


same  form,  the  exciting  current  and  magnetization 
curves  varying  to  meet  this  requirement.  When  the 
pressure  curve  is  peaked,  the  magnetization  curve  will 
be  flat  and  not  reach  so  high  a  maximum  as  for  an 
equivalent  sine  curve.  When  the  pressure  curve  is  flat 
the  magnetization  curve  will  be  peaked  (Sect.  77).  Since 
the  iron  loss  of  a  transformer  depends  upon  the  maxi- 
mum value  of  the  magnetism,  it  is  to  be  expected  that 
a  peaked  pressure  curve  will  cause  a  minimum  iron  loss 
in  a  transformer.  Steinmetz  gives  the  appended  table 
in  support  of  this  : 

HYSTERESIS   LOSS  IN  WATTS. 


Transformer  Number. 

Sine  Wave. 

Peaked  Wave. 

Difference. 

35992 

42.3 

38.6 

9-8   % 

36067 

4I.I 

37-8 

8-95% 

36668 

36.8 

33-9 

8-7   % 

35799 

36.6 

33-7 

8-7   % 

Steinmetz  also  gives  another  case  where  the  loss  in 
a  '200  K.W.  transformer  was  13  per  cent  less  when 
worked  on  a  distorted  curve  than  when  a  sine  curve  was 
impressed.  The  change  in  efficiency  is,  however,  very 
slight.  Experience  has  shown  that  alternating  current 
arc  lights  give  the  best  results. when  worked  on  a  flat- 
topped  curve,  while  Ryan,  Duncan,  and  others  claim 
that  a  sine  curve  gives  the  best  efficiency  in  the  opera- 
tion of  induction  motors.*  The  latter  is  shown  to  be 


*  Electrical  World,  Vol.  24,  pp.  107,  155,  and  178. 


DESIGN   OF   TRANSFORMERS.  519 

theoretically  correct  in  Section  188,  and  if  the  deduction 
proves'to  be  of  much  importance  in  practice,  an  approxi- 
mate sine  form  will  probably  give  the  best  general 
satisfaction.  Roessler  has  lately  shown  that  the  iron 
loss  of  a  transformer  depends  upon  the  ratio  of  the  mean 
value  of  the  pressure  wave  to  its  effective  value. 

136.   Example    of    Transformer    Calculations.  -  -  The 

,          ,      „.       V^-n-n'Nf.  .     . 

formula  E'  = g — —  in   any  practical  case  contains 

only  two  unknown  quantities,  and  these  may  be  suitably 
chosen  by  a  designer  to  suit  his  conditions.  Thus, 
suppose  it  be  desired  to  design  a  10  :  i  transformer  of 
1650  watts  capacity,  for  a  primary  pressure  of  noo 
volts  and  a  frequency  of  125.  This  is  equivalent  to  a 
1500  watt  transformer  which  is  rated  on  the  usual  basis 
of  1000  volts.  By  solving  the  equation,  the  product  of 
n'Nis  given,  thus 

io8  E' 
n'N=  — = —  =  200,000,000  approximately. 

V27T/" 

Calculation  of  Copper  and  Core  Dimensions.  - 
Assuming  the  number  of  turns  in  the  primary  coil  to 
be  650,  makes  the  maximum  magnetization  N=  308,000. 
Now  taking  the  maximum  magnetic  density,  ^  =  3000, 
gives  the  core  cross-section  102.7  sq.  cm.  The  forms 
taken  by  the  cores  and  coils  of  transformers  which  are 
intended  for  ordinary  single-phase  work  are  quite  lim- 
ited, and  the  more  important  ones  are  shown  in  Fig.  236. 
Choosing  whichever  one  of  these  lends  itself  to  the 
requirements  (in  this  case,  say,  the  first)  the  dimensions 
of  the  core  may  be  quickly  determined.  The  apertures 


520 


ALTERNATING   CURRENTS. 


\ 


No.  1 


No.  3 


No.  2 


No.  4 


Fig.  236 


DESIGN   OF   TRANSFORMERS.  521 


No.  5 


No.  6 


No.  7 


CORE 


522 


ALTERNATING   CURRENTS. 


in  the  stampings  must  be  sufficiently  large  to  admit  the 
windings.  The  primary  coil  contains  650  turns  of  a 
wire  having  a  cross-section  of  about  2500  circular  mils, 
allowing  1500  circular  mils  per  ampere,  which  gives  a 
No.  16  B.  &  S.  wire.  This  has  a  diameter  when  double 
cotton  covered  of  about  67  mils,  and  consequently  the 
primary  coil  occupies  a  space  of  about  2,920,000  square 
mils.  The  secondary  winding  contains  65  turns  of  wire 
having  a  cross-section  of  about  22,500  circular  mils, 
allowing  the  same  current  density  as  in  the  primary 
coil,  which  gives  a  No.  7  B.  &  S.  wire.  This  has  a 


Fig.  237 

diameter  of  about  160  mils  when  double  cotton  covered, 
and  the  secondary  coil  occupies  about  1,670,000  square 
mils.  The  total  space  occupied  by  coils  is,  therefore, 
about  4,600,000  square  mils.  This  would  make  a  win- 
dow 2\  inches  square.  Thirteen  wires  of  160  mils 
diameter  go  into  this  space,  so  that  the  secondary  coil 
may  be  wound  in  5  layers  of  13  turns  each  (Fig.  237). 
It  is  usual  to  wind  transformer  coils  on  forms,  and  thor- 
oughly insulate  them  before  the  stampings  composing 
the  core  are  put  in  place.  The  insulation  consists  of 
mica,  micanite,  mica  cloth,  and  varnished  canvas. 


DESIGN   OF   TRANSFORMERS.  523 

Allowing  |  inch  for  insulation,  the  secondary  coil  will 
occupy  a  space  2.2  inches  by  .92  inches.  The  primary 
coil  will,  under  equal  conditions,  wind  in  20  layers 
of  31  turns  each,  and  one  layer  of  30  turns;  and  the 
space  occupied  is  2.2  inches  by  1.53  inches.  The  total 
space  occupied  by  the  coils  is,  therefore,  2.20  inches  by 
2.45  inches,  and  allowing  about  ^  of  an  inch  for  extra 
insulation  and  clearance,  the  apertures  in  the  stampings 
must  be  2-1  inches  by  2|  inches.  The  width  of  the 

o  J  o 

tongue  between  the  apertures  depends  upon  the  length 
of  the  transformer.  It  is  desirable  to  make  the  length 
of  the  path  of  the  magnetic  lines  of  force  as  short  as 
possible,  and  it  is  also  desirable  to  make  the  length  of 
copper  as  short  as  possible,  both  on  the  score  of  regula- 
tion and  cost.  Consequently,  the  length  of  the  trans- 
former should  not  be  too  great.  If  the  tongue  be 
made  2|-  inches  wide  (6.35  cm.),  the  length  of  the  iron 
in  the  transformer  becomes  6.4  inches.  To  make  up 
this  length  requires  512  stampings  .0125  inch  thick. 
The  cores  of  transformers  are  usually  made  of  thin 
stampings  without  special  insulation  between,  the  oxide 
or  a  little  varnish  being  relied  upon  as  sufficient ;  and 
at  least  90  per  cent  of  the  total  length  of  the  trans- 
former may  therefore  be  considered  as  made  of  iron. 
This  makes  the  total  length  of  the  transformer  under 
consideration  7^-  inches.  The  dimensions  of  the  core 
are  shown  in  Fig.  238.  It  would  perhaps  be  better  in 
the  majority  of  cases  to  make  the  apertures  in  the 
stampings  \  inch  larger  each  way,  and  thus  allow  addi- 
tional clearance  and  insulation. 

Calculation  of  Hysteresis  Loss.  —  In   order  to  deter- 


524 


ALTERNATING   CURRENTS 


mine  whether  this  transformer  will  serve  its  purpose, 
it  is  necessary  to  calculate  the  core  loss,  the  copper 
loss,  the  magnetic  leakage,  and  also  the  radiating  sur- 
face per  watt.  The  hysteresis  loss  in  the  core  is 
readily  calculated  by  the  usual  hysteresis  formula 
(Vol.  I.,  p.  74).  The  number  of  cubic  centimeters  of 
iron  in  the  core  is  3900.  This  gives  a  loss  by  hys- 


9% 


Fig-.  238 

teresis  at  a  frequency  of  125  of  37.5  watts  for  iron  in 
which  the  hysteresis  constant  is  21  x  io~n  based  on  the 
cubic  centimeter  and  the  cycle  per  second.  This  value 
is  35  x  io~13  when  based  on  the  cubic  centimeter  and 
cycle  per  minute,  as  the  constants  are  given  on  page  75 
of  Vol.  I.  That  21  x  io~n  is  a  fair  average  value  for 
the  hysteresis  constant  of  the  best  transformer  iron 
is  shown  by  a  series  of  tests  made  by  Steinmetz,  the 


DESIGN   OF   TRANSFORMERS.  525 

average  of  which  gives  17  x  icr11.*  Where  a  series  of 
transformers  is  to  be  designed  with  the  same  kind 
of  iron  in  the  core,  it  is  advantageous  to  plot  a  curve 
which  gives  the  hysteresis  loss  in  watts  per  cubic 
centimeter,  or  per  pound,  of  iron  at  different  magnetic 
densities.  Such  a  curve  for  a  very  good  quality  of  iron 
(hysteresis  constant  =  21  x  io~n)  is  shown  in  Fig.  239 
for  a  frequency  of  100.  The  loss  in  the  iron  when  sub- 
jected to  other  frequencies  may  be  determined  by  mul- 
tiplying the  values  shown  by  the  ratio  J-.  On  account 

of  the  differences  in  the  quality  of  iron  it  is  never  safe 
to  depend  upon  curves  drawn  from  tests  of  one  brand 
to  represent  the  quality  of  another  brand,  and  in  fact 
some  transformer  iron  shows  losses  not  much  more  than 
two-thirds  as  great  as  those  indicated  in  Fig.  239 ;  but  a 
better  grade  than  that  shown  cannot  be  uniformly 
obtained  in  the  market. 

Hysteresis  testing  may  be  carried  out  on  such  instru- 
ments as  those  of  Professor  Ewing  f  or  Mr.  H olden, J  or 
it  may  be  carried  out  by  punching  the  sample  into  trans- 
former plates,  inserting  them  in  a  coil,  and  measuring 
the  losses  by  a  wattmeter.  The  value  of  Bm  may  be 
determined  from  the  number  of  turns  in  the  coil  and  the 
readings  of  a  voltmeter  at  its  terminals.  For  the  re- 
sults to  be  absolute,  the  wave  of  impressed  pressure 
must  approximate  closely  to  a  sinusoid,  but  satisfactory 
comparative  results  may  be  had  independently  of  the 

*  Trans,  Amer.  Inst.  E.  E.,  Vol.  n,  p.  705. 

t  Jour.  Inst.  E.  E.,  Vol.  24,  p.  398;   London  Electrician,  Vol.  34,  p.  786, 

\  Electrical  World,  Vol.  25,  p.  687, 


526 


ALTERNATING   CURRENTS. 

001  W     ''WO  'HO  H3d  S-LLVM 

11 


32       §8        ^      1s- 

ooi  —  /  '-an  a  3d  SJ.J.VM 


DESIGN   OF   TRANSFORMERS.  527 

form  of  the  pressure  wave,  provided  the  same  form  of 
wave  is  used  in  all  tests. 

Calculation  of  Foucault  Current  Loss.  —  The  exact 
calculation  of  the  foucault  current  loss  in  the  core  of  a 
transformer  is  more  difficult  than  that  of  the  hysteresis 
loss  ;  but  equal  exactness  is  not  essential  since  the 
foucault  current  loss  very  seldom  exceeds  25  per  cent 
of  the  core  losses  and  often  constitutes  only  from  10 
to  15  per  cent. 

A  formula  is  developed  by  Mr.  Steinmetz*  which  for 
laminated  iron  is  u  —  (dfB^  io~16,  where  u  is  the  watts 
lost  per  cubic  centimeter  of  iron,  d  is  the  thickness  of 
the  plates  in  mils,  f  the  frequency  of  magnetic  rever- 
sals, and  Bm  the  maximum  magnetic  density.  The  total 
loss  in  watts  for  M  cubic  centimeters  of  iron  is 


Reducing  this  to  the  loss  per  cubic  inch  of  iron  simply 
introduces  the  constant  coefficient  16.4  into  the  for- 
mula. Reducing  the  formula  to  the  watts  lost  per 
pound  of  iron  changes  the  constant  to  nearly 

U'  = 


A  similar  formula  is  deduced  by  Thomson  and 
Ewing.f 

For  the  transformer  under  consideration,  iron  12.5 
mils  thick  is  used  and  the  foucault  current  loss  is 

3900  x  (12.5  x  125  x  30oo)2  x  io~16  =  8.5  watts. 


*  Trans.  Amer.  Inst.  E.  £.,  Vol.  II,  p.  600. 

t  London  Electrician ,  Vol.  28,  p.  631;   Weekes'  Alternating  Current 
Transformers,  p.  2U 


528 


ALTERNATING   CURRENTS. 

WATTS  PER  CD.  CEN. /=  1OO 


o  o  o  o  o 

«  t-  g  §  HJ  CO 

*,  WATTS  PER  LB.  /=10O 


DESIGN   OF   TRANSFORMERS.  529 

The  total  core  loss  for  the  transformer  is  therefore 
46.0  watts  or  2.8  per  cent,  which  is  a  little  high  for  a 
transformer  of  this  size,  and  the  number  of  turns  in  the 
coils  might  be  increased  and  the  cross-section  of  the 
iron  proportionally  decreased  with  advantage.  The 
thickness  of  iron  which  is  here  taken  is  a  usual  one, 
though  transformer  stampings  are  made  of  various  thick- 
nesses between  10  and  18  mils.  The  commonest  thick- 
nesses are  12.5  and  15  mils.  It  is  possible  to  plot  a 
curve  of  the  losses  due  to  foucault  currents  in  sheets 
of  fixed  thickness  similar  to  that  already  given  for  hys- 
teresis, or  a  curve  may  be  plotted  which  gives  the  com- 
bined losses  which  are  found  in  stampings  of  a  given 
thickness  and  made  of  a  fixed  quality  of  iron.  Figure 
239*2  shows  a  curve  for  the  calculated  foucault  cur- 
rent loss  in  stampings  of  the  three  thicknesses,  10,  15, 
and  20  mils,  at  a  frequency  of  100.  The  loss  at  any 
other  frequency  may  be  found  by  multiplying  the  values 

/    f  \2 

from  the  curves  by    *-=•—    . 
J  \iooj 

Calculation  of  Copper  Loss. — The  copper  losses  of 
the  transformer  come  out  as  follows :  The  mean 
length  of  a  turn  is  practically  30  inches.  The  second- 
ary coil  has  65  turns  or  165  feet  of  No.  7  wire,  which 
makes  the  cold  resistance  .08  ohms ;  allowing  a  rise  of 
50°  C.  in  temperature  gives  at  full  load  a  loss  of  21.6 
watts.  The  primary  coil  consists  of  650  turns,  or  1650 
feet  of  No.  16  wire,  which  makes  the  cold  resistance 
6.35  ohms.  The  copper  loss  at  full  load  with  the  trans- 
former hot  is  therefore  practically  40  watts  or  2.42  per 
cent,  which  is  reasonable. 

2M 


530  ALTERNATING   CURRENTS. 

Calculation  of  Exciting  Current.  —  The  final  ques- 
tion to  be  determined  in  relation  to  the  transformer 
is  the  exciting  current.  This  is  made  up  of  two  factors 
(Sect.  113)  which  are  in  quadrature,  i.e.  the  magnetiz- 
ing component  and  the  loss  component.  The  latter  is 
evidently  equal  to  the  core  loss  in  watts  divided  by  the 

primary  pressure,  or  -4 —  =  .042,  and  is  in   step  with 

1 100 

the  primary  pressure.     The  magnetizing  component  C ^ 
is  found  from  the  formula 


or 


when  /  is  the  length  of  the  lines  of  force  in  the  core  and 
A  the  area,  which  gives  for  C M  in  the  case  under  con- 
sideration, 

3000  x  37.5  113,000 

c  „  =  —     — -^ —  — •  =  .049. 

J-75  x  650  x  2000      2,300,000 

The  total  exciting  current  is  therefore, 

C{  =  ^.049*  +  .042"  =  .064, 

which  is  a  satisfactory  value.  The  value  //.  =  2000, 
which  is  here  used,  is  a  fair  value  for  transformer  iron. 

The  value  of  the  magnetizing  component  is  here 
calculated  without  considering  the  effect  of  the  joints 
in  the  magnetic  circuit  upon  its  reluctance.  This  is 
impossible  to  estimate,  and  it  may  cause  a  considerable 
increase  in  the  exciting  current.  Its  effect  may  be 
experimentally  determined  from  measurements  made 


DESIGN   OF   TRANSFORMERS.  531 

on  one  transformer,  and  the  correction  applied  in 
designing  other  transformers  of  the  same  type. 

Arrangement  of  Conductors. — The  primary  conduct- 
ors of  transformers,  except  those  of  very  large  size, 
are  usually  of  wire,  and  the  secondary  conductors  are 
of  wire  or  ribbon.  Wires  which  are  larger  than  No.  7 
or  No.  8  B.  &  S.  gauge  are  not  often  used,  and  con- 
ductors of  larger  cross-section  are  made  of  single 
ribbons,  or  of  two  or  more  wires  or  ribbons  in  parallel. 
The  thickness  of  ribbon  conductor  which  is  used 
seldom  exceeds  75  mils,  since  the  insulation  of  thicker 
conductors  is  likely  to  be  injured  in  winding  around  the 
small  radii  at  the  ends  of  the  coils.  The  insulation 
commonly  consists  of  two  or  three  cotton  coverings 
exactly  as  in  armature  windings,  and  the  finished  coils 
are  very  thoroughly  insulated  with  rubber  tape,  mica, 
etc.  Sometimes  tape  or  cord  wrappings  are  used  on 
ribbon  conductors  instead  of  cotton  coverings,  and  bare 
ribbons  may  be  wound  up  with  tape  between ;  but  the 
latter  is  not  advisable.  The  primary  coil  should  always 
be  broken  into  sections  by  inserting  oiled  paper  or 
varnished  muslin  between  the  layers,  and  where  the 
coils  are  divided  for  the  purpose  of  sandwiching  to 
reduce  magnetic  leakage,  the  primary  coil  should  have 
the  least  number  of  divisions  so  that  its  insulation  may 
be  less  interfered  with. 

Checks. — The  total  radiating  surface  from  which 
heat  may  leave  this  transformer  is  nearly  400  square 
inches,  and  the  total  losses  at  full  load  make  86  watts, 
or  we  have  nearly  five  square  inches  per  watt  lost. 
The  efficiency  of  the  transformer  at  full,  half,  and 


532  ALTERNATING   CURRENTS. 

quarter  load,  is  95.0  per  cent,  93.6  per  cent,  89.4  per 
cent.  The  all-day  efficiency,  based  on  five  hours  full 
load  and  19  hours  no  load,  is  86.4  per  cent.  The  num- 
ber of  turns  per  volt  in  the  windings  is  —  in  this 

VOutput 

transformer.  In  transformers  of  the  best  makes  in- 
tended for  frequencies  between  60  and  135,  the  number 
of  turns  per  volt  in  the  windings  seems  to  vary  between 

—     •*         and  —  >  when    the   output   is  given   in 

VOutput          VOutput 

watts,  and  the  numerator  seems  usually  to  be  less  than 

25  for  the  best  transformers.  This  ratio  gives  a  guide  to 
the  choice  of  the  number  of  turns  which  shall  be  used 
in  a  transformer,  and  the  number  of  lines  of  force  in  the 
magnetic  circuit  is  then  fixed  by  the  formula.  In  deter- 
mining the  size  of  the  plates  and  the  length  of  the  core, 
the  ratio  of  the  over-all  area  of  the  plates  to  the  area  of 
the  apertures  may  be  used  as  a  guide.  In  the  above 
example  this  ratio  is  4.00,  and  in  a  large  number  of 
commercial  transformers  of  capacities  from  500  watts 
to  30,000  watts  the  ratio  is  found  to  vary  from  2.75  to 
4.25,  with  an  average  of  about  3.00.  A  final  check 
upon  any  design  may  be  made  to  depend  upon  the 
calculated  core  loss  per  pound  or  per  cubic  centimeter 
of  iron,  which  varies  in  commercial  transformers  from 
.80  to  2.80  watts  per  pound,  or  .012  to  .042  watts  per 
cubic  centimeter,  and  averages  in  first-class  trans- 
formers about  i.oo  watt  per  pound,  or  .015  watts  per 
cubic  centimeter.  In  any  case,  the  determination  of 
the  most  economical  design  in  a  particular  form 
depends  upon  working  out  several  designs  with  dif- 
ferent constants.  The  best  design  may  then  be  chosen. 


DESIGN  OF  TRANSFORMERS. 


533 


137.  Dimensions     of     Various     Commercial     Trans- 
formers. —  The   data    for    1 500   watt    transformers    of 
various  manufacturers  are  given  on  page  534  for  com- 
parison.    All  the  data  are  based  on  a  primary  pressure 
of   iioo  volts,  frequency  of  125,  and  ratio  of  transfor- 
mation of  10 :  i. 

138.  Calculation  of  Magnetic  Leakage.  —  In  the  pre- 
ceding example  no  attempt  has  been  made  to  calculate 
the  magnetic  leakage.     By  properly  placing   the   coils 
with  respect  to  each  other  and  to  the  magnetic  circuit, 
the  magnetic   leakage  may  be   made   almost    or   quite 


Fig.  240 

negligible.  Thus  in  Fig.  240  the  coils  are  so  placed 
that  a  short-circuiting  of  lines  of  force  along  the  path 
indicated  by  the  dotted  lines  is  to  be  expected  ;  but  when 
the  coils  are  arranged  as  shown  in  Fig.  241,  the  leakage 
is  not  likely  to  be  great ;  while  if  the  coils  are  divided  and 
the  parts  sandwiched  together,  the  leakage  may  be  made 
very  small.  The  leakage  may  be  calculated  quite  ap- 
proximately by  the  method  indicated  below.  Since  the 
currents  in  the  primary  and  secondary  coils  of  the 
transformer  are  in  practical  opposition  of  phase,  their 
magnetizing  effects  are  opposite.  This  tends  to  cause 


534 


ALTERNATING   CURRENTS. 


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DESIGN    OF   TRANSFORMERS. 


535 


lines  of  force  to  short-circuit  through  the  coils,  as 
shown  in  Fig.  240,  the  tendency  being  greatest  at  the 
plane  where  the  coils  touch  each  other,  and  falling 


Fig.  241 

off  to  zero  at  the  outer  edges  of  the  coils,  so  that  the 
magnetic  leakage  will  differ  for  each  layer  of  wire  in 
the  coils.  The  effect  of  leakage  must  therefore  be  cal- 
culated for  each  layer,  and  the  total  effect  may  then  be 


ALi. 


summed  up.  In  Fig.  242  the  ordinates  of  the  line  A'BA" 
are  proportional  to  the  ampere-turns  acting  at  any  point 
to  cause  leakage  lines  to  pass  through  the  coils.  These 


536 


ALTERNATING  CURRENTS. 


ordinates  are  equal  to  the  number  of  turns  in  the  coils 
between  the  foot  of  the  ordinate  and  the  outer  edge  of 
the  coils,  multiplied  by  the  current  flowing  in  the  turns. 
The  ordinate  is  evidently  zero  at  the  outer  edges  of 
the  coils,  and  a  maximum  equal  to  practically  n" Cn  at 
the  plane  between  the  coils.  The  number  of  the  leak- 
age lines  of  force  enclosed  by  any  layer  is  proportional 


T 


a-mn 


Fig.  243 


to   the    corresponding    ordinate    of    the    lines    CDC". 
These  ordinates  are  respectively  equal  to 


x  — 


where  x  is  the  desired  ordinate,  y  is  the  mean  ordinate 
of  the  line  A'BA"  taken  from  the  neutral  plane  at  D  to 
the  point  under  consideration,  a  is  the  area  of  a  coil  be- 
tween the  neutral  plane  and  the  point  under  consid- 
eration, and  /  is  the  length  of  the  lines  of  force  through 


DESIGN   OF   TRANSFORMERS.  537 

the  coils  (Fig.  243).     The  maximum  value  of  x  falls  at 
the  outer  edges  of  the  coils  and  is 


_ 


where  Al  is  the  total  iron  surface  presented  to  a  coil 
from  which  leakage  lines  emerge,  and  the  average  num- 
ber of  leakage  lines  enclosed  by  the  different  layers  at 
the  instant  of  maximum  leakage  is 

x^  _  i.2$n"C"A1  _  . 
~~ 


The  inductive  effect  of  this  leakage  on  the  secondary 
winding  is  equal  to 


io8 

and  an  equivalent  effect  is  produced  on  the  secondary 
on  account  of  the  leakage  of  lines  of  force  through  the 
primary  coil.  If  A  be  taken  to  represent  the  total  area 
of  iron  from  which  leakage  lines  emerge,  which  is  pre- 
sented to  both  coils,  the  formulas  become 

Ar      i.2Sn"C"A 

and    Et  = 


t  t 

•X/2/  10 

The  inductive  effect  due  to  leakage  is,  as  has  already 
been  shown  (Sect.  112),  to  be  in  quadrature  with  the- 
active  pressure  in  the  secondary  circuit,  En  .  Conse- 
quently, the  drop  in  secondary  pressure  caused  by  mag- 
netic leakage  is 

£»-^  £»*-£*, 


538         ALTERNATING  CURRENTS. 

and  the  total  drop  of  secondary  pressure  in  the  trans- 
former is 


E"  -  (\/E"*  -  E?  +  C'R"  +  -  C'R'). 

The  value  for  El  in   the  transformer  designed    above, 
when  the  coils  are  wound  one  inside  of  the  other,  is 

r      4  x  652x  15  x  218  x  125  . 

El  =  -       *          o-      —^=io  volts; 

6.7  x  io8 

when  the  coils  are  placed  side  by  side,  this  becomes 

_      4  x  652  x  15  x  242  x  125 

Et  =  -  ^  -=12  volts. 

6.0  x  io8 

The  leakage  drop  is,  therefore, 

100  —  Vioooo  —  100=    .5  volt 


or  100  —  Vioooo  —  144  =  .72  volts. 

This  may  be  made  practically  negligible  with  either  ar- 
rangement of  the  coils  by  dividing  one  of  the  windings 
into  two  parts  and  sandwiching  the  other  between  them. 
This  reduces  N%  to  one-fourth  and  the  leakage  drop  in  a 
still  greater  ratio. 

139.  Joints  in  the  Magnetic  Circuit.  —  In  this  dis- 
cussion no  account  has  been  taken  of  the  position  of 
the  joints  in  the  stampings,  which  really  have  a  marked 
effect  on  both  the  magnetic  leakage  and  exciting  current. 
Every  effort  is  made  to  reduce  the  magnetic  reluctance 
of  the  joints  by  making  them  as  few  in  number  as  possi- 
ble, and  arranging  them  so  that  the  joints  of  adjoining 
plates  come  in  different  positions  in  the  core.  The 
joints  in  the  magnetic  circuit  are  usually  one  or  two, 


DESIGN   OF   TRANSFORMERS.  539 

and  should  never  be  more  than  two.  Lapping  joints 
are  really  essential  to  the  best  performance,  and  while 
butt  joints  have  been  used  in  transformers,  their  effect 
has  always  been  detrimental.  The  arrangement  of 
joints  is  shown  very  plainly  in  Fig.  236.  The  last  four 
views  shown  are  of  antiquated  forms.  The  arrange- 
ment of  the  joints  may  be  made  in  various  ways  with 
each  form  of  built-up  core,  as  the  position  of  the  joints 
depends  only  on  the  punching. 

140.  Ageing  of  Transformer  Cores.  —  Experience  has 
shown  that  the  core  loss  in  some  transformers  increases 
to  a  very  considerable  extent  during  the  first  few 
months  of  operation.  The  increase  in  loss  is  due  to  an 
increased  hysteresis  loss  per  cycle,  and  was  originally 
ascribed  by  Ewing*  to  magnetic  fatigue  of  the  iron. 
It  has,  however,  been  quite  conclusively  proved  by  ex- 
periments f  and  the  records  of  transformer  manufact- 
urers to  be  caused  by  the  continuous  condition  of  high 
temperature  at  which  the  iron  is  operated.  The  ageing 
seems'  to  have  the  greatest  effect  upon  poor  qualities  ot 
iron,  hastily  and  imperfectly  annealed,  and  the  least 
effect  upon  the  best  grades  of  iron  which  have  been 
annealed  with  great  care.  The  conditions  under  which 
the  annealing  of  the  transformer  plates  is  performed, 
especially  with  reference  to  temperature  and  duration 
of  the  process,  have  much  to  do  with  the  extent  of  the 
ageing  effect,  and  by  proper  annealing  it  can  be  ren- 
dered very  small  in  cores  made  of  proper  qualities  of 
iron.  The  iron  now  generally  used  for  transformer 

*  London  Electrician,  Vol.  34,  p.  161. 
t  Ibid.,  pp.  160,  190,  191,  219,  297,  498. 


540         ALTERNATING  CURRENTS. 

cores  is  a  very  mild  steel  made  by  either  the  bessemer 
or  open-hearth  processes,  though  puddled  iron  sheets 
are  still  used  to  some  extent. 

141.  Current  Rushes.  —  It  has  been  shown  in  Section 
25  that  the  exponential  term  in  the  complete  equation 
for  an  alternating  current  in  an  inductive  circuit  is  ordi- 
narily negligible,  but  under  certain  conditions  its  effect 
for  a  few  periods  after  the  current  is  started  in  a  circuit 
may  be  considerable.  This  question  was  investigated  by 
Fleming  *  and  others  f  with  especial  reference  to  the 
action  of  transformers  when  first  switched  onto  an 
alternating-current  circuit.  If  a  transformer  is  switched 
onto  a  circuit,  the  current  does  not  instantly  assume  the 
final  form  of  the  wave,  but  rises  gradually  through  a 
short  interval  to  its  final  form.  The  length  of  the  inter- 
val and  the  magnitude  of  the  early  current  depends 
upon  the  reactance  of  the  circuit,  the  frequency,  and 
the  point  in  the  pressure  wave  at  which  the  connection 
is  made.  If  the  instant  of  switching  onto  the  circuit  is 
that  at  which  the  impressed  pressure  is  passing  through 
zero  the  current  in  the  transformer  is  less  during  the 
early  interval  than  its  final  value,  while  if,  at  the  instant 
of  switching  on,  the  impressed  pressure  is  passing 
through  its  maximum  value  there  may  be  quite  an  ex- 
cess of  current  flow  through  the  circuit  for  a  short  time, 
on  account  of  the  relations  which  exist  between  the 
instantaneous  impressed  and  counter  electric  pressures 
during  the  first  half  period.  The  abnormal  state  of  the 

*  Jour.  Inst.  E.  £.,  Vol.  21,  p.  677. 

t  Hay,  On  Impulsive  Current-rushes  in  Inductive  Circuits,  London 
Electrician,  Vol.  33,  pp.  229,  277,  and  305. 


DESIGN   OF   TRANSFORMERS. 


541 


current  can  only  exist  for  a  very  short  time  unless  the 
reactance  of  the  circuit  approaches  a  condition  of  res- 
onance (Appendix  D),  which  is  very  exceptional.  As. 
far  as  the  operation  under  ordinary  conditions  of  trans- 
formers or  other  commercial  alternating-current  appli- 
ances is  concerned,  the  phenomena  of  current  rushes 
may  be  entirely  neglected. 

142.  Impedance  Coils,  Compensators,  etc.  —  The  design 
of  impedance  coils,  reactance  coils,  choking  coils,  or  econ- 


1OO  VOLTS 


COIL. 
95  VOLTS 


VARC 
|S3O  VOLTS 


,  LINE 


ARC 


Fig.  244 


omy  coils  as  they  are  variously  called,  is  carried  out  very 
much  in  the  same  manner  as  the  design  of  a  trans- 
former. These  coils  consist  of  a  magnetic  circuit  with  a 
winding  of  small  resistance,  but  large  inductance.  The 
magnetic  circuit  and  winding  are  proportioned  in  exactly 
the  same  manner  as  the  primary  winding  of  a  trans- 
former, using  the  formula  E  = 7rn0  m*  -  Coils  of  this 

io8 

type  are  used  for  a  variety  of  purposes  where  it  is 
desired  to  throttle  the  flow  of  current  without  the 
attendant  loss  of  power  which  always  follows  the  use  of 
resistances.  Where  arc  lamps  are  used  on  constant- 
pressure  alternating-current  circuits,  an  economy  coil 


542  ALTERNATING   CURRENTS. 

is  ordinarily  used  to  reduce  the  pressure  from  the  50 
or  100  volts  on  the  wires  to  the  30  volts  required  at 
the  lamp  (Fig.  244).  The  pressures  upon  the  distribut- 


Fig.  245 


ing  circuits  in  a  theatre,  or  upon  the  feeders  of  a  plant 
furnishing  alternating  currents  for  incandescent  light- 
ing, may  be  regulated  by  impedance  coils.  Figure  245 
shows  a  regulator  or  Booster  intended  for  this  purpose, 


Fig.  246 


which  may  be  used  either  to  reduce  the  pressure  in 
the  circuit  or  to  raise  it,  and  which  is  not  a  true  im- 
pedance coil,  as  its  effect  is  due  to  transformer  action. 


DESIGN   OF   TRANSFORMERS.  543 

Numerous  devices  of  this  kind,  which  depend  upon 
moving  the  primary  and  secondary  coils  with  reference 
to  each  other  or  the  core  with  reference  to  both,  have 
been  manufactured.  Figure  246  shows  an  impedance 
coil  adapted  to  an  incandescent  lamp  socket  which  is 
arranged  so  that  by  turning  the  key  the  number  of 
turns  of  the  winding  included  in  the  circuit  is  varied, 
and  the  lamp  is  thus  turned  up  and  down.  Figure  247 
shows  the  Thomson  impedance  coil,  in  which  the  reac- 


Fig.  247 

tive  effect  is  varied  by  moving  a  heavy  copper  shield 
so  as  to  more  or  less  enclose  the  winding,  instead  of 
varying  the  number  of  turns  of  the  winding  included 
in  the  circuit.  This  shield  acts  like  the  short-circuited 
secondary  of  a  transformer,  and  therefore  reduces  the 
apparent  impedance  of  the  windings  as  it  approaches 
them. 

There  is  another  type  of  inductive  apparatus  which  is 
little  used  and  which  goes  under  the  name  of  Compensa- 
tor. This  consists  of  a  single  winding  on  a  proper  mag- 


544 


ALTERNATING   CURRENTS. 


netic  circuit,  to  which  both  the  primary  and  secondary 
circuits  are  connected.  Figure  248  shows  the  connec- 
tions of  a  220  volt  compensator  which  feeds  two  1 10  volt 
secondary  circuits.  In  this  case  the  function  of  the  com- 


Fig.  248 


pensator  is  to  equalize  the  pressure  between  the  two 
secondary  circuits  regardless  of  their  relative  loads. 
This  purpose  is  fulfilled  fairly  well,  but  the  regulation 


T 


Pig.  249 


is  not  as  satisfactory  as  that  of  a  transformer.  Figure 
249  shows  the  connections  of  a  220  volt  compensator 
which  supplies  a  1000  volt  secondary  circuit.  When 
one  of  the  secondary  terminals  of  a  compensator 


is 


DESIGN   OF   TRANSFORMERS.  545 

arranged  so  that  the  position  of  its  connection  to  the 
winding  may  be  varied,  the  machine  is  called  an  Auto- 
transformer.  The  secondary  pressure  and  current  of  an 
auto-transformer  may  be  arranged  to  vary  through  any 
desired  range,  while  the  primary  current  changes  only 
so  far  as  is  required  by  any  change  in  the  power  ab- 
sorbed by  the  secondary  circuit. 

2N 


546  ALTERNATING   CURRENTS. 


CHAPTER    XIII. 

POLYPHASE    CONDUCTING    SYSTEMS    AND    THE    MEASURE- 
MENT   OF    POWER    IN    POLYPHASE    CIRCUITS. 

143,  Polyphase  Conducting  Systems.  —  A  full  discus- 
sion of  conducting  systems  has  no  place  in  this  book, 
but  a  brief  explanation  of  the  methods  of  connecting 
the  coils  of  polyphase  machines  and  the  wires  of  poly- 
phase circuits  is  essential  for  the  purposes  of  the  follow- 
ing chapters. 

Polyphase  systems  are  usually  operated  with  either 
two  currents  with  approximately  90°  difference  of  phase, 
or  three  currents  with  approximately  120°  phase  differ- 
ence. Polyphase  machines  arranged  for  two  currents 
are  called  Two-phase  Machines,  or  Two-phasers.  Those 
arranged  for  three  currents  are  called  Three-phase  or 
Tri-phase  Machines,  Three-phasers  or  Tri-phasers  (see 
page  386). 

The  transmission  circuits  for  two-phase  currents 
may  be  arranged  to  be  entirely  independent  of  each 
other,  four  wires  being  then  required  (Fig.  250)  ;  or, 
three  wires  may  be  used,  in  which  case  one  of  them 
is  common  to  the  two  currents  (Fig.  251)  ;  the  current 
in  the  third  or  common  wire,  at  any  instant,  is  equal  to 
the  algebraic  sum  of  the  currents  in  the  other  two,  and 
the  algebraic  sum  of  the  instantaneous  currents  in  the 


POLYPHASE    CONDUCTING    SYSTEMS. 


547 


three  wires  is  always  equal  to  zero.  The  effective  cur- 
rent in  the  common  return  wire  is  equal  to  the  vector 
sum  of  the  two  circuit  currents;  and  is,  therefore,  VTC, 
where  C  is  the  effective  current  in  one  circuit,  pro- 
vided the  currents  are  equal  in  the  two  circuits  and 
have  a  phase  difference  of  90°,  which  is  the  condition 


b' 


Fig.  25Oa 


Fig.  250 

when  the  system  is  properly  designed  and  symmetri- 
cally loaded  or  Balanced.  The  pressure  between  the 
two  outside  wires  of  the  two-phase  system  with  com- 
mon return  is  the  vector  sum  of  the  two  circuit  press- 
ures, and  is,  therefore,  V2  E  in  a  balanced  system, 
where  E  is  the  pressure  between  one  side  and  the 
common  return.  The  com- 
mon current  and  pressure  in 
a  balanced  system  are  45° 
from  the  phase  of  the  cur- 
rent in  either  of  the  inde- 
pendent wires.  Figure  252 
shows  the  graphical  compo- 
sition of  the  pressures.  A 
and  B  are  the  two  line  press-  Fig.  251 


548 


ALTERNATING   CURRENTS. 


ures,  and  R  is  the  resultant  pressure  measured  across 
the  outside  wires. 

The  coils  of  two-phase  machines  may  be  entirely 
independent  of  each  other,  in  which  case  four  collec- 
tor rings  are  required,  or  the  circuits  may  be  joined  so 
as  to  require  only  three  collector  rings.  In  some  two- 
phase  machines  the  armature  is  wound  with  the  equiva- 
lent of  a  series-path  continuous-current  winding,  and 
four  collector  rings  and  independent  circuits  are  then  re- 


Fig.  252 


quired  to  avoid  short-circuiting  portions  of  the  armature. 
Figure  253  shows,  by  diagram,  various  ways  of  connect- 
ing the  coils  of  two-phase  machines  (see  Sect.  102  a). 

It  is  possible,  in  three-phase  systems,  to  use  three 
entirely  independent  circuits,  each  consisting  of  two 
wires,  and  carrying  currents  of  120°  difference  of  phase; 
but  in  practice  the  circuits  are  almost  invariably  combined 
so  as  to  use  three  wires,  and  the  current  in  each  wire 
is  then  equal  to  the  vector  sum  of  two  circuit  currents. 


POLYPHASE   CONDUCTING   SYSTEMS. 


549 


The  coils  of  three-phase  machines  may  be  connected 
together  so  that  they  form  the  three  sides  of  a  triangle 


Fig.  253 

with  the  transmission  wires  connected  to  the  three  cor- 
ners of  the  triangle  (Fig.   254),  or  one  end  of  each  coil 

A 


Fig-.  254 

may  be  individually  connected  to  the  transmission  wires, 


550 


ALTERNATING   CURRENTS. 


the  free  ends  of  the  coils  being  connected  together  (Fig. 
255).*    In  either  case  the  number  of  transmission  wires 


Fig.  255 

is  three,  and  the  algebraic  sum  of  their  instantaneous  cur- 
rents is  always  eqiial  to  zero.  In  the  latter  or  Star 
arrangement,  which  is  often  represented  by  the  symbol 

R 


Fig.  256 

Y,  the  pressure  between  any  two  line  wires  in  a  balanced 
system  is  V^  E,  where  E  is  the  pressure  in  one  coil  of 
the  machine.  Thus  in  Fig.  256,  if  a,  b,  and  c  are  the  press- 


*  See  Section  iO2<z. 


POLYPHASE   CONDUCTING   SYSTEMS. 


551 


ures  in  a  vector  diagram,  the  pressure  between  A  and  C 
is  OR,  which  is  A/3  E  in  magnitude  and  has  30°  differ- 
ence of  phase  from  a  or  —  c.  In  Fig.  257  the  curve  R 
shows  the  potential  difference  between  A  and  C.  The 
line  current  must  in  this  case  be  the  same  as  that  pass- 
ing through  the  coil  to  which  it  is  attached.  In  the 
Triangle  or  Mesh  winding,  which  is  often  represented 
by  the  symbol  A,  the  pressure  between  wires  is  evi- 


B' 


Pig.  257  : 

dently  that  generated  by  one  coil,  and  the  current  in  the 
line  wire  is  the  resultant  of  that  in  two  adjacent  coils, 
or  A/3  C  in  a  balanced  system,  where  C  is  the  current  in 
a  coil.  This  may  be  obtained  from  Fig.  256  by  consid- 
ering a,  b,  and  c  to  be  currents.  If  a  is  the  initial  cur- 
rent from  which  the  phase  is  measured,  b  must  be  taken 
backward,  as  this  current  must  flow  in  the  opposite 
direction  from  a,  in  order  to  get  to  the  line  which  is 


552 


ALTERNATING   CURRENTS. 


connected  to  the  junction  of  a  and  b.  This  makes  the 
resultant  current  30°  from  the  current  a.  Figure  258 
shows  various  ways  in  which  the  coils  of  three-phase 
machines  may  be  connected.  The  arrangements  are 
either  of  the  star  or  mesh  connection  or  a  combination 
thereof. 


Fig.  258 


144.  Uniform  Power  in  Polyphase  Systems. —  In  gen- 
eral, the  power  transferred  in  a  balanced  polyphase 
circuit  is  uniform  throughout  each  period,  and  the 
torque  exerted  by  balanced  polyphase  machinery  is 
uniform.  This  is  different  from  the  conditions  in 
single-phase  circuits,  where  the  power  has  been  shown 
to  vary  from  a  maximum  to  a  minimum  during  every 
quarter  period  (Sect.  43).  In  the  case  of  a  single-phase 
circuit  the  power  at  any  instant  is  cmem  sin  (a  —  (/>)  sin  a 


POLYPHASE   CONDUCTING   SYSTEMS.  553 

=  cmem  sin2  a  cos  c/>  —  cmem  sin  a  cos  a  sin  $,  which  varies 
with  a.  In  a  balanced  two-phase  circuit  the  instan- 
taneous power  is  cmcm  sin2  a  cos  <f>  -f  £m<?TO  sin2  (a  —  90°) 
cos  c/>  =  cmem  cos  c/>  (sin2  a  -f  cos2a)  =  ^OT<?m  cos  <£,  which  is 
constant.  In  the  same  way  the  power  in  a  balanced 
three-phase  circuit  is  cmen  cos  <J>  {sin2 a  +  sin2  (a  —  120°) 
+  sin2  (a  —  240°) }  =  %cmem  cos  <£,  which  is  constant ;  and, 
in  general,  the  power  in  any  balanced  polyphase  circuit 

in  which  the  phase  differences  are  equal  to  —  or  — , 

m         m 

where  m  is  the  even  or  odd  number  of  phases,  is, 

,     (       .     9  .     J  27T\  .     „/  4?T\    , 

cmem  cos  cf>  j  sin2  a  +  sm2f  a J  +  siin  a  —  -1—  j  -f  •  •  • 

.   „/        2(m—  I)TT\  ) 

+  sin(a-  ^-J|' 


which  is  equal  to  f  —  J  cmemcos<f),  and  is  constant,  since 

.     o  .     9/  2  7T\  .     _/  4  7T\ 

sm2a  +  sin2  a +  sin2  a  —  -^— )+  ••• 

V  «r  /  V          m  / 


.   9/         2  (m  —  I)TT\ 

>in2(  a -    = 

V  m         J 


The  uniformity  of  power  in  a  balanced  polyphase 
circuit  may  also  be  directly  deduced  from  the  proposi- 
tion that  the  resultant  of  m  equal  harmonic  motions 

acting  in  lines  having  an  angular  difference  of  — —  is  a 

m 

uniform  circular  motion  with  an  amplitude  equal  to  — 
times  the  amplitude  of  the  components. 

*  Todhunter's  Plane  Trigonometry,  p.  243. 


$54  ALTERNATING   CURRENTS. 

145.  Algebraic  Sum  of  Instantaneous  Currents  is  Zero. 
—  It  is  also  easily  proved  that  the  algebraic  sum  of  the 
instantaneous  currents  in  a  balanced  polyphase  circuit 
of  any  number  of  phases,  m,  is  always  equal  to  zero. 
Thus 


.-/  (m-  I)TT\ 

cm  =  cmax  sin  a  —  2-  -  —  )• 

\  m        J 

Hence,  C1  +  cz  +  c3-]  -----  h  cm=cmax  \  sin  a  +  sin  (a  —  —  \ 


+  sinq  -  4g 


-  2 


\         m  V  m 


but  evidently,   sin  a  +  sinf  a  —  —  ]  +  sinf  a  —  ^  ]  H 

\         m)  \         mj 


(m-i)7T\_ 


m 


=  o, 


and  therefore  cl  +  c%  +  cB-\  -----  h  cn  =  o.  When  poly- 
phase circuits  have  an  odd  number  of  phases,  the 
number  of  line  wires  may  be  equal  to  the  number  of 
phases,  but  when  the  number  of  phases  is  even,  the 
number  of  line  wires  must  be  one  greater  than  the 
number  of  phases,  f 

*  Todhunter's  Plane  Trigonometry,  p.  243. 

t  Blondel,  Elementary  Theory  of  Rotary  Field  Apparatus,  La  Lumiere 
£lectrique,  Vol.  50,  p.  351. 


MEASUREMENT   OF   POWER.  555 

146.   Relations    between    Currents   and   Pressures.  — 

The  following  are  the  relations  between  the  currents 
and  the  pressures  in  the  lines  and  coils  of  a  balanced 
three-phase  system  developed  from  the  earlier  discus- 
sion (Sect.  143) : 

1 .  Star  Connection.     Cl=Cc\  EAB  =  EAC = EBG  =  V 3 Ec. 
Line  pressure  EAB  is  the  vector  sum  of  coil  pressures  Ea 
and  Eb  and  is  30°  behind  the  phase  of  coil  pressure  Eb. 
Line  pressure  EAG  is  the  vector  sum  of  coil  pressures  Ea 
and  Ec  and  is  30°  behind  the  phase  of  coil  pressure  Ea. 
Similar  relations  hold  for  the  other  two  corners. 

2.  Mesh    Connections.     EAB  =  E&  Cl  =  V3  Cc.     Line 
current  CA  is  the  vector  sum  of  coil  currents  Ca  and  Cb 
and  is  30°  ahead  of  the  phase  of  coil  current  Ca  and  30° 
behind  the  phase  of  coil  current  Cb.      Similar  relations 
hold  for  the  other  corners. 

The  subscripts  applied  to  the  letters  C  and  E  in  the 
paragraphs  above  have  the  following  meanings :  /,  line ; 
c,  coil ;  a,  b,  specific  coils ;  A,  B,  specific  lines ;  AB, 
AC)  BC,  measurements  between  the  respective  corners 
of  the  circuits.  Figures  254,  255,  256,  and  257  should 
be  used  for  reference. 

If  the  circuits  of  utilization  in  a  polyphase  system 
are  not  machines  (for  instance  incandescent  lamps),  the 
devices  must  be  connected  exactly  as  would  be  the  coils 
of  a  machine,  unless  transformers  intervene,  in  which 
case  the  secondary  circuits  may  be  independent;  but  the 
load  should  be  uniformly  distributed  to  keep  the  system 
balanced.  In  three-phase  circuits  in  which  the  gener- 
ator coils  are  connected  in  star  fashion,  a  fourth  wire 
may  be  introduced  which  runs  from  a  common  junction 


556  ALTERNATING   CURRENTS. 

of  the  three  branches  of  the  load  to  the  neutral  point  of 
the  generator,  but  this  method  is  not  used  commercially. 

147.  Effect  of  Mutual-  and  Self-Induction  between  Cir- 
cuits. —  The    effects   of   self-  and  mutual-induction    in 
polyphase    circuits    may  be   determined   by    using   the 
principles   already  set  forth  (Sects.  47    and   no),  pro- 
vided the   resultant   effect   of   the    differing   phases  is 
always   properly  considered.      In    unbalanced    systems 
the  mutually  inductive  influence  of  the  circuits  tends  to 
increase  their  defects  in  balance.*     In  order  to  regulate 
phase    pressures    independently,    pressure     regulators 
such  as  those  described  in  Section  82,  or  rheostats,  must 
be    introduced    in    each    phase ;    or,    if    the    generator 
armature  is  stationary,  the  number  of   active   conduc- 
tors on   each  phase  may  be  varied  by  a  commutator. 
(Example :    Stanley   alternator.)      In    some    polyphase 
alternators  where  the  armatures  of  the  different  phases 
are  influenced  by  different  field  frames,  the  regulation 
may  be  effected  by  varying  the  field  magnetism  (Ex- 
ample :    large   Westinghouse   alternators),    but  this   is 
an  unusual  construction. 

148.  Measurement  of  Power  in  Two-  and  Three-phase 
Circuits.  —  The  principles  underlying  the  methods  of 
measuring    power   in    polyphase   circuits   differ   in   no 
respect  from  those  already  deduced  in  relation  to  single- 
phase   circuits  (Sect.  44),  but   it  is  desirable  to  apply 
them   in    such    a    way    as    to   reduce    the    number    of 
necessary  readings   to  a   minimum.      For   satisfactory 
measurements,  non-inductive  wattmeters  are  of   essen- 
tial    importance,    and     very    satisfactory    commercial 

*  Electrical  World,  Vol.  25,  p.  302. 


MEASUREMENT   OF   POWER. 


557 


portable  wattmeters  are  now  to  be  had  for  a  reason- 
able price. 

A.    Two- Phase  Systems. 

A  i.  Independent  Circuits.  In  a  two-phase  system 
with  separate  circuits,  independent  wattmeter  readings 
are  taken  in  each  circuit  and  the  total  power  is  the  sum 


Fig.  259 

of  the  readings.  One  wattmeter  placed  in  each  circuit 
(Fig.  259),  from  which  simultaneous  readings  are  taken, 
is  the  best  arrangement;  but  if  two  wattmeters  are  not 
to  be  had,  one  may  be  inserted  successively  in  the  two 
circuits,  and  the  sum  of  the  readings  is  equal  to  the 
power  in  the  system,  provided  the  load  does  not  vary 
while  the  readings  are  being  taken.  If  the  circuit  is 
perfectly  balanced,  twice  the  reading  of  a  wattmeter  in 


558 


ALTERNATING   CURRENTS. 


Fig.  260 


Fig.  26Oa 


MEASUREMENT   OF   POWER. 


559 


one  circuit  is  equal  to  the  power,  but  this  is  a  condition 
which  cannot  be  relied  upon. 

A  2.  Circuits  with  Common  Return. — Two  wattmeters 
may  here  be  used,  one  for  each  circuit,  connected  in 
the  way  shown  in  Fig.  260.  The  arrangement  shown 
in  Fig.  260  a  is  equivalent  to  a  single  wattmeter  con- 
nected as  in  Fig.  261,  and  is  only  correct  for  a  system  in 


Fig.  261 

exact  balance.  When  the  single  wattmeter  is  used  in  a 
balanced  system,  the  current  coil  is  placed  in  the  com- 
mon wire,  and  a  reading  is  .taken  with  the  free  end  of 
the  pressure  coil  connected  to  one  outside  wire.  The 
pressure  coil  terminal  is  then  quickly  transferred  to  the 
other  outside  wire  and  a  new  reading  taken.  The  con- 
dition of  exact  balance  is  not  to  be  relied  upon,  so  that 
the  arrangement  of  Fig.  260  must  ordinarily  be  used. 


56o 


ALTERNATING   CURRENTS. 


The  sum  of  the  readings  of  the  two  wattmeters  then 
gives  the  power  in  the  system. 

B.    Three-Phase  Systems. 

B  i.  Three  Wattmeters. — a.  If  the  power  delivered 
by  a  generator,  or  absorbed  by  a  motor  or  other  device, 
which  is  connected  in  star  fashion,  is  to  be  measured, 
three  wattmeters  may  be  used,  connected  as  shown  in 


Fig.  262 

Fig.  262,  provided  the  common  or  neutral  point  is  acces- 
sible. It  is  evident  that  each  wattmeter  measures  the 
power  in  one  coil  so  that  the  sum  of  the  readings  gives 
the  power  in  the  system.  If  the  system  is  exactly 
balanced,  three  times  the  reading  of  one  wattmeter 
gives  'the  power. 

b.  If  the  devices  are  connected  mesh  fashion,  three 
wattmeters  may  still  be  used,  provided  the  current  coils 
of  the  wattmeters  can  be  inserted  directly  into  the  coil 


MEASUREMENT   OF   POWER. 


56l 


circuits  as  shown  in  Fig.  263.  The  power  in  the  circuit 
is  equal  to  the  sum  of  the  three  wattmeter  readings, 
and  if  the  circuit  is  exactly  balanced,  three  times  the 
reading  of  one  wattmeter  gives  the  power. 

c.  When  it  is  impossible  to  insert  the  wattmeters  in 
the  coil  circuits  of  a  device  with  mesh  connection,  the 
three-wattmeter  method  may  still  be  used  by  the  crea- 
tion of  an  artificial  neutral  point  as  shown  in  Fig.  264. 
For  this  purpose,  three  equal  non-reactive  resistances 
A 


Fig.  263 

are  connected  together  at  one  end,  and  the  other  ends 
are  connected  to  the  respective  corners  of  the  mesh 
circuit.  The  pressure  between  the  neutral  point  and 

J7 

either  corner  is  equal  to  — =,  where  E  is  the  pressure 

Vs 

of  one  coil,  and  the  phase  of  this  pressure  is  <£  degrees 
in  advance  of  the  current  entering  the  corner.  A  watt- 
meter with  its  current  coil  inserted  in  the  circuit  wire 

20 


562 


ALTERNATING   CURRENTS. 


leading  to  the  corner  carries  a  current  equal  to  V^  C, 
and  if  the  free  end  of  the  pressure  coil  is  connected  to 
the  neutral  point,  the  power  reading  of  the  wattmeter  is 


which  is  the  power  in  the  coil.      Care  must  be  taken 
that  the  resistances  of  the  wattmeter  pressure  coils  are 


W 


Fig.  264 


so  large  compared  with  the  three  auxiliary  resistances 
that  connecting  them  in  circuit  does  not  disturb  the 
pressure  of  the  neutral  point.  If  the  resistances  of  the 
wattmeter  pressure  coils  are  exactly  equal,  auxiliary 
resistances  are  unnecessary,  and  the  measurement  may 
be  made  by  joining  the  free  ends  of  the  three  pressure 
coils. 

These  methods  are  independent  of  the  condition  of 
balance  in  the  system  or  the  current  lag. 


MEASUREMENT   OF   POWER. 


563 


B  2.  Two  Wattmeters. — The  algebraic  sum  of  the 
readings  of  two  wattmeters,  inserted  in  a  three-phase 
circuit  as  shown  in  Fig.  265,  gives  the  power  in  the 
system  with  entire  independence  of  the  balance  of  the 
system  or  current  lag.  When  the  current  lag  in  the 
circuit  is  less  than  60°,  or  the  power  factor  is  greater 
than  .50,  the  arithmetical  sum  of  the  readings  is  equal 
to  the  power  in  the  circuit;  but  if  the  lag  is  greater 


W 


W 


Fig.  265 


than  60°  (the  power  factor  is  less  than  .50),  the  rela- 
tion of  the  currents  in  the  current  and  pressure  coils 
of  one  of  the  wattmeters  causes  it  to  have  a  negative 
reading,  and  the  arithmetical  difference  of  the  read- 
ings of  the  two  instruments  gives  the  power.  There 
is  some  difficulty  in  distinguishing  which  condition 
exists  in  many  cases,  especially  when  the  power  ab- 
sorbed by  partially  loaded  induction  motors,  in  which 
the  power  factor  is  low,  is  under  measurement.  As 


564  ALTERNATING   CURRENTS. 

a  general  rule,  if  the  conditions  do  not  make  the  case 
evident,  the  truth  may  be  discovered  by  interchanging 
the  positions  of  the  instruments  without  altering  the 
relative  connections  of  their  main  and  pressure  coils. 
If  the  deflections  of  both  needles  are  reversed,  the  dif- 
ference of  the  original  readings  represents  the  power, 
but  if  the  deflections  are  in  the  same  direction  as 
before,  the  sum  of  the  readings  is  correct.  The  proof 
of  this  theorem  is  given  in  Section  149. 

A  double  wattmeter,  consisting  of  two  fixed  coils  and 
two  movable  coils  on  one  spindle,  can  be  used  in  meas- 
uring power  by  the  two  wattmeter  method.  Such  an 
instrument  of  itself  sums  up  the  double  reading  algebra- 
ically, and  a  single  reading  gives  the  power.  Recording 
wattmeters  based  upon  this  principle  can  be  made  very 
useful  in  the  commercial  sale  of  power  from  two-phase 
and  three-phase  circuits. 

B  3.  One  Wattmeter.  —  In  a  balanced  circuit  one 
wattmeter  may  be  very  conveniently  used  by  connect- 
ing the  current  coil  in  one  wire  and  connecting  the 
free  terminal  of  the  pressure  coil  alternately  to  the 
other  two  leads  (Fig.  266),  when  the  sum  of  the  read- 
ings gives  the  power.  For,  the  power  reading  of  the 
wattmeter  in  its  first  position  is  A/3  CE  cos  ((/>  4-  30°), 
and  in  its  second  position,  V3  CE  cos  (<£  —  30°),  and 
the  sum  of  the  readings, 

A/3  CE{cos  (0  4-  30°)  4-  cos  (<£  -  30°)  \=$CE  cos  fa 

where  C,  E,  and  <£  are  the  current,  pressure,  and  lag 
in  a  coil ;  but  CE  cos  </>  is  the  power  in  one  coil  and 
3  CE  cos  </>  is  the  total  power  of  the  three  coils,  hence 


MEASUREMENT   OF    POWER 


565 


the  one  wattmeter  gives  correct  indications  provided 
<f>  is  the  same  for  all  the  coils,  and  the  load  is  uniformly 
A 


Fig.  266 

distributed.  A  wattmeter  having  two  independent 
pressure  coils  could  be  used  as  a  direct  reading  instru- 
A 


Fig.  266  a 


ment  for  this  purpose.     A  similar  wattmeter  could  also 
be  used    in    one-wattmeter  measurements  of   power  in 


566  ALTERNATING   CURRENTS. 

two-phase  circuits.  An  ordinary  wattmeter  with  one 
pressure  coil  may  be  used  for  the  double  measurement 
at  one  observation  by  connecting  the  free  end  of  the 
pressure  coil  to  the  middle  of  a  high  non-inductive 
resistance  which  connects  the  two  lines  opposite  to  the 
one  in  which  the  current  coil  is  inserted  (Fig.  2660). 
This  reading  is  evidently  equal  to  the  sum  of  the  read- 
ings with  the  other  arrangement,  but  the  wattmeter 
constant  must  be  determined  with  one-half  of  the  high 
resistance  in  series  with  it. 

149.  Measurement  of  Power  in  Any  Polyphase  Circuit.* 
—  In  the  case  of  a  polyphase  system  of  m  phases  and 
m  conductors,  the  power  in  the  circuit  may  be  measured 
by  m  —  i  wattmeters.  Supposing  A,  B,  C,  D,  etc.,  are 
points  where  the  m  conductors  of  a  polyphase  supply 
circuit  connect  to  the  circuits  under  test,  then,  as  has 
been  already  proved  (Sect.  145),  ILc  =  o,  if  c  represents 
the  instantaneous  current  in  any  branch.  The  power 
supplied  through  the  A  conductor  at  any  instant  is 

equal  to  ~*  =  cava,  where  qa  is  the  quantity  of  elec- 
tricity brought  to  A  during  a  time  dt,  and  va  is  the 
absolute  electrical  potential  of  A. 

The  average  power  transferred  through  conductor  A 

i    CT 
during  a  complete  period  is   —  I    cavadt,  and  the  total 

7  c/O 


I    CT 
power  in  the  circuit,    W=^L—\    cvdt. 

/  J 0 


*  Blondel,  Measurement  of  the  Energy  of  Polyphase  Currents,  Proc. 
Elect.  Congress  held  at  Chicago,  p.  112;  Lunt,  Measurement  of  the  Power 
of  Polyphase  Currents,  Electrical  World,  Vol.  23,  p.  771. 


MEASUREMENT   OF   POWER.  567 

The  absolute  potentials  of  the  points  are  inconvenient 
to  measure,  and  it  is  desirable  to  introduce  into  the 
formula  the  difference  between  the  pressures  at  the 
points  and  some  fixed  point  of  potential  vf .  Since 
Z<:  =  o,  we  also  have  ILcv'  =  o,  and  ^Lcv'  may  therefore 
be  directly  inserted  in  the  formula  without  destroying 
the  equality,  or 

W=2±  f  V  -  cv')dt  =  2±C'C(v-  v')dt. 


Writing  e  for  v  —  v'  (the  instantaneous  difference  of 
pressure  between  the  fixed  point  and  any  given  point 
in  the  system)  gives 


The  fixed  point  may  be  taken  at  one  of  the  corners  of 
the  circuit,  A  for  instance,  since  2«/  =  o  holds  equally 
for  it,  and  the  power  formula  becomes 


^  f 

1  «/0 


etc.  + 

1 


or  W=  CbEab  cos  &  +  CcEac  cos  6"  +  etc. 

where  0',  6nt  etc.,  are  the  angular  differences  in  the 
phases  of  the  pressures  and  currents.  The  terms  on 
the  right  of  the  equation  are  the  familiar  forms  rep- 
resenting the  power  readings  of  a  wattmeter,  so  that 
if  m  —  i  wattmeters  are  inserted  in  circuit  with  their 
current  coils  respectively  in  the  m  —  i  conductors,  B, 
C,  D,  etc.,  and  the  free  ends  of  their  pressure  coils 


568  ALTERNATING   CUR-RENTS. 

all  connected  to  A,  the  algebraic  sum  of  their  readings 
is  equal  to  the  power  in  the  system. 

When  the  circuit  includes  three  phases  only,  the 
formula  is  W=  CbEab  cos  0'  +  CcEac  cos  0",  and  only 
two  wattmeters,  connected  as  in  Fig.  265,  are  required 
to  give  the  power  in  the  circuit,  but  due  regard  must  be 
had  to  the  relative  signs  of  cos  0'  and  cos  0".  From 
the  relative  phases  of  the  currents  Cb  and  Ce  and  press- 
ure E&  (Sect.  146)  it  is  easy  to  see  that  in  a  balanced 
system  0'  =  <£'  -  30°,  and  also  that  0"  =  $"  +  30°,  and 
therefore 

W=  C,Eal  cos  (0'  -  30)°  +  CeEac  cos  (<£"  +  30°), 

in  which  $  and  (f)ir  are  the  angles  of  lag  of  the  circuit 
currents.  The  formula  shows  that  the  first  term  at 
the  right,  which  represents  the  reading  of  one  watt- 
meter, is  positive  within  the  limits  <f>'  =  +  90°  and 
<£'  =  —  60°,  and  that  its  value  is  negative  between  the 
limits  </>'  =  —  60°  and  <£'  =  —  90°.  The  second  term, 
which  represents  the  reading  of  the  second  wattmeter, 
is  positive  between  the  limits  —  90°  and  +  60°  and 
negative  between  -f  60°  and  +  90°.  Consequently,  if 
the  current  lags  equally  in  the  circuits,  or  <£'  =  <f>fr,  both 
wattmeters  have  a  positive  reading,  and  the  power  in  the 
circuit  is  the  sum  of  the  readings,  for  angles  of  lag 
between  +  60°  and  —  60°.  If  the  angle  of  lag  is  +  60° 
(the  current  lags  behind  the  pressure),  the  second  watt- 
meter reading  is  zero,  and  the  power  in  the  circtiit  is 
equal  to  the  reading  of  the  first  wattmeter.  If  the  lag  is 
greater  than  +  60°,  the  reading  of  the  first  wattmeter  is 
positive  and  the  second  is  negative,  and  the  power  in  the 


MEASUREMENT   OF   POWER. 


569 


circuit  is  equal  to  the  difference  of  the  two  readings. 
Again,  if  the  angle  of  lag  is  —  60°  (the  current  leads 
the  pressure),  the  first  wattmeter  reading  is  zero,  and 


ANGLE 


OF  LAG 


30C 


30 


\ 


I 


Fig.  267 

the  power  in  the  circuit  is  equal  to  the  reading  of  the  second 
instrument.  If  the  lead  is  more  than  60°,  the  reading  of 
the  first  instrument  is  negative  and  of  the  second  posi- 


570         ALTERNATING  CURRENTS. 

tive,  and  the  power  in  the  circuit  is  equal  to  the  difference 
of  the  two  readings.  When  the  angle  of  lag  is  ±  90°, 
the  readings  of  the  two  instruments  are  equal,  but  one 
is  positive  and  the  other  negative.  The  relation  of  the 
wattmeter  readings  to  the  angle  of  lag  between  +  90° 
and  —  90°  are  shown  by  the  curves  in  Fig.  267.  These 
are  two  equal  sinusoids  with  a  phase  difference  equal  to 
60°.  The  readings  of  two  wattmeters  in  a  balanced 
three-phase  circuit  at  any  value  of  the  lag  are  in  the 
proportion  of  the  corresponding  ordinates  of  the  two 
curves.  The  same  conditions  obtain  in  an  unbalanced 
circuit,  provided  equivalent  angles  of  lag  are  considered. 


ALTERNATING-CURRENT    MOTORS  571 


CHAPTER   XIV. 

ALTERNATING-CURRENT    MOTORS. 

150.     Alternators    as    Synchronous    Motors.*  —  Any 

alternator  may  be  run  as  a  motor,  provided  it  is 
brought  up  to  synchronous  speed  and  into  step  be- 
fore it  is  thrown  into  circuit.  The  motor  will  then 
run  in  complete  synchronism  if  left  to  itself.  If  it  is 
overloaded,  or  by  other  means  is  thrown  out  of  synchro- 
nism, it  will  stop.  In  general,  the  action  of  an  alter- 
nator used  as  a  synchronous  motor  is  quite  similar  to 
that  of  an  alternator  operated  in  parallel  with  another. 
A  great  disadvantage  of  single-phase  synchronous 
motors  is  the  fact  that  they  are  not  self-starting,  but 
must  be  brought  up  to  speed  before  they  will  operate ; 
and  while  polyphase  synchronous  motors  may  be  made 
to  start  themselves  without  load,  the  operation  is  uneco- 
nomical. The  starting  of  single-phasers  may  be  done 
by  a  small  series-wound  auxiliary  motor  made  with  lam- 
inated fields.  Such  a  motor  will  run  when  placed  in 
an  alternating-current  circuit,  since  -the  magnetism  of 
the  fields  and  armature  will  reverse  together  as  the  cur- 

*  Picou,  Transmission  de  Force  par  Moteurs  alternatifs  synchrones, 
Bull.  Soc.  Int.  Electriciens,  Vol.  12,  p.  60 ;  Blondel,  Couplage  et 
Synchronization  des  Alternateurs,  La  Lumiere  Eleclrique,  Vol.  45,  pp. 
425>  563 ;  Rhodes,  A  Theory  of  Synchronous  Motors,  Phil.  Mag.,  July, 
1895;  Alt-  Current  Motors,  Land.  Elect,  fieview,  Vol.  37,  pp.  182,  222. 


5/2 


ALTERNATING   CURRENTS. 


rent  changes  direction;  but  very  little  power  can  be 
developed  by  such  a  machine  on  account  of  its  enor- 
mous self-inductance.  A  small  two-phase  motor  (Sect. 
182),  with  a  device  for  splitting  the  current  into  two 
phases,  may  be  used  (see  Fig.  268);  or  the  exciter  of 


TRANSFORMER 

COMPENSATOR 

VOLTMETER 


Fig-    268 

the  alternator  may  be  run  as  a  motor  by  a  storage 
battery  and  used  to  bring  the  alternator  into  synchro- 
nism, the  storage  battery  being  recharged  by  cur- 
rent from  the  exciter  after  the  alternator  is  operating 
on  the  circuit.  Polyphase  synchronous  motors  may  be 
started  by  an  ordinary  polyphase  induction  motor,  such 
as  is  described  in  later  sections. 


ALTERNATING-CURRENT   MOTORS.  573 

While  the  practical  disadvantage'  of  synchronous 
motors,  due  to  the  fact  that  they  are  not  self-starting, 
may  be  overcome  by  these  special  devices,  the  expense 
of  motor  equipment  is  increased,  and,  at  the  best,  the 
motor  cannot  be  started  under  load.  Consequently, 
synchronous  motors  are  not  satisfactory  for  general 
power  distribution.  They  have  been  used  with  con- 
siderable satisfaction  in  certain  special  plants  for  the 
long-distance  transmission  of  power,  and  may  be  said 
to  be  destined  to  play  an  important  part  for  such  work ; 
but  for  general  power  transmission  and  distribution  pur- 
poses, they  cannot  be  satisfactorily  used. 

151,  Relation  of  Field  Strength  to  the  Working  of 
Synchronous  Motors.  —  When  a  synchronous  motor  is 
put  in  the  circuit,  a  peculiar  relation  exists  between  the 
strength  of  the  field  of  the  motor  and  the  current  in 
its  armature.  In  continuous-current  motors,  if  the 
strength  of  the  field  is  slightly  changed  without  alter- 
ing any  of  the  other  conditions,  the  speed  of  the  motor 
changes  inversely,  and  the  current  in  the  armature 
remains  practically  unchanged ;  but  the  speed  of  a 
synchronous  motor  cannot  change  permanently,  and, 
consequently,  upon  first  consideration,  it  would  appear 
that  the  field  of  a  synchronous  motor  must  be  exactly 
adjusted,  in  order  that  the  machine  may  operate  sat- 
isfactorily. This,  however,  is  proven  not  to  be  the 
case  in  practice,  on  account  of  the  effect  which  may 
be  gained  through  variations  of  the  relative  phases 
of  the  current  and  of  the  impressed  and  counter 
pressures.  The  active  pressure,  which  at  any  instant 
causes  current  to  flow  through  the  armature  of  a 


574  ALTERNATING   CURRENTS. 

motor,  is  equal  to  the  difference  of  the  correspond- 
ing instantaneous  values  of  the  impressed  and  counter 
pressures.  If  the  field  strength  of  a  motor  is  so 
adjusted  that  the  values  of  the  impressed  and  coun- 
ter pressures  are  equal,  and  the  motor  armature  is 
brought  into  exact  step  with  the  impressed  pressure 
curve,  then,  when  the  motor  is  switched  on  the  supply 
main,  it  will  fall  back  in  phase  with  respect  to  the 
impressed  pressure,  sufficiently  to  permit  the  proper 
load  current  to  pass  through  the  armature.  Now  sup- 
pose that  at  some  instant  the  load  is  increased,  the 
difference  of  instantaneous  pressures  at  that  instant 
will  be  insufficient  to  pass  the  current,  which  is  neces- 
sary for  the  new  load,  through  the  armature.  The 
motor,  therefore,  falls  back  in  its  phase  without  losing 
synchronism,  if  the  load  is  not  too  great,  and  then 
continues  operating  in  synchronism,  but  with  a  greater 
lag  in  step.  When  a  motor  lags  in  step  behind  the 
phase  of  impressed  electromotive  force,  its  counter 
pressure  lags  to  an  equal  extent.  The  armature  cur- 
rent ordinarily  takes  an  intermediate  phase,  so  that  it 
is  behind  the  resultant  pressure,  but  in  advance  of 
opposition  to  the  counter  pressure. 

Were  it  not  for  the  effect  of  the  current  lag  with 
respect  to  the  resultant  pressure,  caused  by  self-induc- 
tance, it  would  be  necessary  to  adjust  the  field  exci- 
tation of  a  synchronous  motor,  so  that  its  counter 
pressure  would  be  less  than  the  impressed  pressure, 
and  the  range  of  load  carried  with  a  given  excitation 
would  be  small.  The  effects  due  to  the  current  lag, 
however,  make  it  possible  to  adjust  the  field  excitation 


ALTERNATING-CURRENT   MOTORS  575 

once  for  all,  so  that  the  motor  may  be  operated  on  a 
widely  varying  load.  It  is  even  possible,  on  account 
of  the  automatic  adjustment  of  the  pressure  phases,  to 
operate  a  motor  when  its  excitation  is  much  greater  or 
much  less  than  its  normal  value.  The  adjustment  is 
assisted  by  the  effect  of  armature  reactions  on  the 
motor,  in  which  a  lagging  current  tends  to  strengthen 
the  fields  and  a  leading  current  to  weaken  them  (Sect. 
70).  When  a  single  motor  is  operated  from  an  alter- 
nator of  about  its  own  size,  the  automatic  adjustment 
of  the  machines  is  still  more  marked,  since  the  current 
which  strengthens  the  field  of  the  motor  tends  to 
weaken  that  of  the  alternator  as  the  load  is  varied, 
and  vice  versa,  which  is  desired.* 

It  is  evident  from  the  preceding  that  the  armature 
current  of  a  motor  must  have  a  wattless  component 
which  depends  directly  upon  the  phase  differences  of 
the  impressed  and  counter  pressures  and  the  angle  of 
lag,  and  it  may  readily  be  seen  that  the  most  economical 
excitation  of  a  synchronous  motor  field  is  that  which 
reduces  the  armature  current  to  a  minimum  (or  makes 
the  power  factor  a  maximum)  when  the  motor  is  carry- 
ing the  average  load. 

152.  Graphical  Illustrations  showing  the  Relations 
of  Pressure  and  Current  in  a  Synchronous  Motor  Arma- 
ture. —  In  order  to  bring  out  more  clearly  the  facts  just 
given,  recourse  may  be  had  to  a  diagram  in  which  rela- 

*  Compare  Ryan,  The  Behaviour  of  Single-Phase  Synchronous  Motors, 
Sibley  Journal  of  Engineering,  May,  1894;  Scott,  Long-Distance  Trans- 
mission for  Electric  Lighting  and  Power,  Trans.  Amer.  Inst,  Elect.  Eng., 
Vol.  9,  p.  425. 


5/6 


ALTERNATING   CURRENTS. 


tions  of  current  and  pressures  are  shown  much  as  in 
parallel  working.  It  was  shown  (Sect.  87)  that  in 
parallel  working  the  machines  were  held  in  step  by  a 
motor  action,  and  that  if  one  machine  was  cut  off  from 
its  prime  mover  it  would  continue  to  run  in  synchronism 
as  a  motor,  its  pressure  being  unchanged.  In  Fig.  269 
let  OC  be  the  current  passing  through  two  alternators, 
one  acting  as  a  motor,  and  let  OL  be  the  pressure  of 
self-induction  (27rfLC\  and  OS  the  active  pressure 


--L 
Fig.  269 

(CR)\  then  OR  will  be  the  resultant  pressure  required 
to  pass  the  current  OC  through  the  circuit.  Suppose 
the  alternator  generates  a  pressure  OEV  and  the  motor 
is  excited  to  give  an  equal  pressure  OE^ ;  then  OE^  and 
OE2  must  be  in  such  a  phase  as  to  give  the  resultant 
OR,  while  the  elements  of  pressure  resolved  upon  the 
current  line  OC  must  be  such  that  the  element  of  OE^ 
has  the  same  direction  as  the  current  and  the  element 
of  OE2  is  in  opposition.  The  work  delivered  by  the 
generator  is  OC  x  OE1cos(/>1,  and  that  utilized  by  the 


ALTERNATING-CURRENT   MOTORS. 


577 


motor  armature  in  furnishing  power  and  overcoming  the 
magnetic  and  friction  losses  is  OC  x  OE^  cos  </>2 ;  while 
that  lost,  due  to  resistance,  is  their  difference,  and  is 

equal  to 

OC  x  OR  cos  </>  =  OS  x  OC. 

It  will  be  seen  that  for  small  loads  the  current  may 
lead  the  generator  pressure  as  shown  in  Fig.  269, 
but  that  as  the  load  increases  (and  the  length  of  OR 

y 


C  SB 


Fig.  269  a 

therefore  increases),  the  generator  pressure  is  caused 
to  swing  forward  so  that  the  current  takes  a  lagging 
position,  as  shown  in  Fig.  269  a.  The  construction 
indicates  that  the  current  is  always  in  the  lead  of  direct 
opposition  to  the  counter  pressure  when  the  impressed 
and  counter  pressures  are  equal,  and  that  the  value  of 
<£>2  increases  when  the  load  on  the  motor  is  increased. 
The  value  of  C  is  one-half  greater  in  Fig.  269*2  than 

2P 


578         ALTERNATING  CURRENTS. 

in  Fig.  269,  the  input  of  the  motor  is  50  per  cent 
greater,  and  the  output  about  45  per  cent  greater. 
The  latter  is  increased  in  a  smaller  proportion  because 
the  C2R  loss  increases  directly  as  C2,  while  the  output 
increases  less  rapidly  than  C .  The  motor  will  continue 
to  operate,  as  the  load  is  increased,  until  <f>2  has  attained 
such  a  value  that  E^  cos  </>2  x  C  becomes  a  maximum ; 
then,  if  an  additional  load  is  put  on  the  motor,  the 
corresponding  increase  of  </>2  will  cause  CE^  cos  02  to 
decrease,  and  the  motor  will  fall  out  of  synchronism 
and  stop,  because  the  maximum  value  of  its  torque  is 
not  sufficient  to  pull  the  load.  In  the  case  under  con- 
sideration (when  the  impressed  and  counter  pressures 
are  equal)  this  will  not  occur  with  well-designed  alterna- 
tors until  a  load  much  above  the  normal  is  reached. 

153.  Impressed  and  Counter  Pressures  Unequal.  —  As 
was  stated  in  a  preceding  section  (Sect.  151),  the  press- 
ure at  which  the  motor  is  run,  and  therefore  its  exci- 
tation, has  an  important  bearing  upon  the  stability  of 
operation  and  the  efficiency  of  transmission.  If  the 
motor  pressure  is  made  larger  than  that  of  the  gener- 
ator, the  current  and  pressure  relations  may  be  shown 
by  a  construction  similar  to  that  used  in  Fig.  269.  Let 
OR  in  Fig.  270  a  represent  the  magnitude  and  direction 
of  the  resultant  pressure,  and  the  impressed  and  counter 
pressures  have  magnitudes  OE^  and  OE^ ;  then  the 
parallelogram  can  be  completed  with  but  one  value 
of  the  angles  ^  and  c/>2,  i.e.  that  shown  in  the  figure. 
In  this  case  the  counter  pressure  is  greater  than  the 
impressed  pressure.  Now  suppose  the  counter  pressure 
is  made  OE\,  having  the  same  horizontal  projection  as 


ALTERNATING-CURRENT   MOTORS. 


579 


while  OR  and  the  impressed  pressure  have  the 
same  values  as  before ;  then  the  phase  relations  are 
as  shown  by  the  lines  OR,  OE^,  and  OE2'.  The  values 
of  OE2  cos  (£2  and  OE2'  cos  fa'  are  equal  by  the  con- 
struction, since  the  points  E2  and  E%  are  in  the  same 
vertical  line,  and  since  OR  has  the  same  magnitude  and 


Fig.  27Oa 

position  in  the  two  cases  the  current  is  the  same,  and 
OEl  cos  fa  and  OE^  cos  fa'  are  equal ;  but  in  the  first 
case  the  counter  pressure  is  greater  than  the  impressed 
pressure  and  the  current  leads  the  impressed  pressure, 
and  in  the  second  case  the  impressed  pressure  is  greater 
and  the  current  lags.  This  construction  shows  that  for 
every  load  on  the  motor,  except  that  corresponding  to  a 


580         ALTERNATING  CURRENTS. 

zero  angle  of  lag,  there  are  two  values  of  the  excita- 
tion which  cause  the  same  armature  current  to  flow, 
the  current  leading  the  impressed  pressure  with  one  ex- 
citation and  lagging  by  an  equal  angle  with  the  other 
excitation ;  hence  an  over-excited  motor  acts  upon  the 
line  current  very  much  like  a  condenser,  and  an  under- 
excited  motor  acts  like  an  inductance  coil.  When  the 
counter  pressure  is  much  less  than  the  impressed  press- 
ure, the  current  may  lag  with  respect  to  opposition  to 
the  counter  pressure,  but  under  no  other  conditions,  and 
this  is  not  a  practical  condition. 

154.  Excitation  which  gives  Greatest  Power  Factor.— 
The  excitation  at  which  the  motor  will  do  the  most  work 
with  a  given  current  flowing,  or  will  carry  a  given  load 
with  the  least  current  (and  hence  do  it  most  efficiently), 
is  that  which  causes  the  current  to  come  into  phase  with 
the  impressed  pressure.  In  this  case  the  current  for  a 
given  load  is  a  minimum,  and  E^  cos  c/>2  is  a  maximum. 
If  the  value  of  O£2  in  Fig.  270  a  is  increased  (by  increas- 
ing the  excitation  of  the  motor),  either  the  length  of  OR 
which  is  proportional  to  C,  or  the  impressed  pressure, 
must  be  increased,  provided  CE^  cos  <£2,  which  is  equal 
to  the  motor  load,  is  constant.  On  the  other  hand,  if 
OE^  decreases  while  OEl  remains  constant,  the  angle 
of  lag,  (f)v  and  the  current,  decrease  until  the  current 
and  impressed  pressure  come  into  phase,  after  which  a 
further  decrease  of  OE^  causes  the  current  to  increase 
again,  as  is  shown  by  the  relations  between  Ely  C,  and 
£%  in  the  figure.  The  value  of  CE^  cos  <£2  is  propor- 
tional to  the  area  of  the  rectangle  Olqp,  since  Ol  is  pro- 
portional to  C.  Now  if  the  motor  load  is  kept  constant, 


ALTERNATING-CURRENT   MOTORS.  581 

but  its  excitation  is  changed,  the  corner  of  the  corre- 
sponding rectangle  must  be  different  from  q,  but  the 
locus  of  the  motion  of  the  corner  is  a  hyperbola  with  its 
origin  at  O  and  the  rectangular  axes  x  and  y,  since  the 
rectangles  included  between  the  axes  and  the  ordinates 
and  abscissas  of  all  points  must  be  of  equal  area.  Con- 
sequently, the  point  of  the  vector  representing  E^  at  the 
least  current  for  a  given  load  must  be  in  the  rectangular 
hyperbola,  mm,  which  passes  through  q.  This  point, 
which  is  E2"  for  the  given  load,  is  found  by  laying  off 
the  horizontal  line  equal  in  length  to  OE^  which  will 
just  reach  between  the  hyperbola  and  the  line  OR. 


This  cuts  OR  at  R't  and  C  =  2  r/£  '  which  is  the 
minimum  current  for  the  load.  The  impressed  pressure 
and  current  are,  under  these  conditions,  in  phase  with 
each  other.  At  this  point  E^1  sin  $%"=  E"b.  If  a  larger  or 
smaller  pressure  than  OE^  is  used,  such  as  OE2  or  OE2r, 
the  impressed  pressure  and  current  are  thrown  out  of 
phase,  and  the  current  in  the  circuit  is  therefore  in- 
creased, which  causes  increased  C2R  losses  and  arma- 
ture reactions.  The  excitation  of  the  motor  which 
brings  the  current  and  impressed  pressure  into  step, 
depends  upon  the  relation  of  the  impedance  of  the 
motor  circuit  to  its  resistance.  If  /  =  2  R,  the  counter 
pressure  is  equal  to  the  impressed  pressure  at  zero 
angle  of  lag,  and  if  1^2  R,  the  counter  pressure  is 
respectively  greater  or  less  than  the  impressed  press- 
ure at  zero  lag.*  The  expression  /=  2R  is  equivalent 

*  Steinmetz,  Theory  of  the  Synchronous  Motor,   Trans.  Amer.  Inst. 
E.E.,  Vol.  u,j>.  767. 


582 


ALTERNATING   CURRENTS. 


to,  reactance  equals  V^R(I  =  ^RZ  +  4  Tr2/2/,2  =  2  R, 
and,  therefore,  2  nrfL  =  ~V$R).  In  the  machines  which 
are  now  commonly  built,  the  impedance  of  the  armature 
circuit  is  commonly  equal  to  or  larger  than  twice  the 
resistance,  so  that  a  maximum  power  factor  is  gained 
in  such  synchronous  motors  by  an  excitation  which 
gives  a  counter  pressure  that  is  equal  to  or  greater 
than  the  impressed  pressure. 

155.    Curve  showing  the  Relation  of  Armature  Current 
to  Excitation.  —  We  may  plot  the  relation  of  armature 


20 
18 
16 
11 
2    12 

te. 

i 

°10 

Ul 

ce. 

i. 

te. 

< 

c 

4 
« 

X 

) 

\ 

>< 

\ 

/ 

\ 

/ 

/ 

\ 

\ 

/ 

\- 

/ 

/ 

\ 

2 

j              8              JA             T>            1*            I6             1{ 
EXCITATION 

Fig.  27Ob 

current  to  field,  excitation  for  a  motor  operating  under 
the  conditions  considered  above,  by  taking  the  corre- 


ALTERNATING-CURRENT   MOTORS.  583 

spending  values  of  E2  and  C  from  a  chart  made  like 
Fig.  270*7.  This  gives  a  curve  like  Fig.  270  b,  which  has 
two  values  of  its  abscissas  for  every  value  of  the  ordi- 
nate  except  the  lowest ;  or  for  each  value  of  the  arma- 
ture current  there  may  be  two  values  of  the  excitation, 
one  being  greater  and  the  other  less  than  the  impressed 
pressure,  except  at  the  point  of  minimum  armature 
current,  which  corresponds  to  but  one  excitation,  as  has 
already  been  explained  (Sect.  153).  The  way  in  which 
Fig.  270*2  is  constructed  shows  that  the  smaller  the  angle 
(f>,  or  the  smaller*  the  armature  self-inductance,  the  less 
will  be  the  difference  in  the  two  excitations  correspond- 
ing to  any  armature  current;  and  hence  the  curve  show- 
ing the  relation  of  excitation  to  current  in  a  machine 
having  a  large  time  constant  is  broad  and  rounded,  but 
the  curve  for  an  armature  having  a  small  time  constant 
is  sharp  and  narrow.  In  an  ideal  machine  without  self- 
inductance,  the  two  values  of  excitation  for  a  leading 
and  lagging  impressed  pressure  are  equal,  and  the  curve 
becomes  a  straight  line. 

156.  Maximum  Load.  —  When  a  synchronous  motor 
is  operated  with  a  fixed  excitation  on  a  variable  load, 
the  step  of  the  motor  will  automatically  adjust  itself, 
when  the  load  changes,  to  the  changed  conditions,  until 
CE<i  cos  ff>2  is  equal  to  the  new  load,  provided  a  certain 
maximum  limit  is  not  exceeded.  As  the  load  increases, 
the  current  must  increase,  whence  it  is  evident  from  Fig. 
270  a  that  the  change  in  phase  of  the  motor  pressure 
must  be  in  the  direction  which  increases  <£2,  and  that 
CE^  cos  c/>2  will  therefore  reach  a  maximum  point  beyond 
which  the  motor  cannot  work,  since  cos  $2  decreases  as 


584         ALTERNATING  CURRENTS. 

</>2  increases.  This  point  depends  upon  the  self-induc- 
tance of  the  armature,  which  controls  <£2,  when  the  im- 
pressed and  counter  pressures  are  constant.  If  the  load 
on  the  motor  is  made  greater  than  the  maximum  value 
of  CE^  cos  </>2,  the  motor  must  fall  out  of  synchronism 
and  stop.  The  smaller  the  impedance  of  the  motor  cir- 
cuit, the  less  will  the  angle  </>2  change  with  any  change 
of  current,  when  E^  remains  constant,  as  is  seen  by  the 
construction  of  Fig.  270  a,  and  therefore  the  maximum 
load  which  the  motor  will  carry  depends  inversely  on  its 
impedance ;  but  the  greater  the  angle  of  lag,  <£,  the  less 
will  be  the  value  of  </>2  for  fixed  values  of  the  pressures 
and  impedance,  and  consequently  the  less  rapidly  will 
cos  <£2  vary  with  a  given  variation  of  c/>2.  So  that  the 
maximum  load  which  a  motor  will  carry  depends  in- 
versely upon  the  impedance  of  its  armature  circuit,  and 
directly  upon  the  angle  by  which  the  current  lags 
behind  the  resultant  pressure.*  In  practice,  armature 
reactions  always  tend  to  weaken  the  field  of  the  motor 
as  the  current  is  in  the  lead  of  opposition  to  the  counter 
pressure;  and  it  is  therefore  advisable  to  excite  the 
machine  rather  above  that  pressure  corresponding  to 
the  least  current  for  normal  load,  as  the  armature  reac- 
tions then  tend  to  decrease  the  field  strength  and  thus 
modify  the  motor  pressure  so  as  to  cause  a  decrease  in  the 
amount  of  armature  current.  If  the  excitation  is  made 
smaller  than  that  corresponding  to  minimum  current, 

*  Mordey,  Alternate-Current  Working,  Jour.  Inst.  E.  £.,  Vol.  18, 
p.  595  ;  Kolben,  Elektrotechnische  Zeitschrift,  Vol.  1 6,  p.  802;  London 
Electrical  Engineer ;  1895;  Kapp's  Electrical  Transmission  of  Energy, 
4th  ed.,  p.  278. 


ALTERNATING-CURRENT    MOTORS. 


585 


the  armature  reactions  cause  the  deviation  from  mini- 
mum current  to  become  still  greater.  To  decrease  the 
effect  of  armature  reactions  as  well  as  make  the  machine 
capable  of  carrying  considerable  overloads  without  being 
dragged  out  of  synchronism,  it  is  advisable  to  use  strong 
fields,  and  armatures  with  the  least  number  of  conductors 
compatible  with  an  economical  design.  There  is  ordi- 
narily no  danger  to  the  motor  if  it  stops,  as  the  arma- 
ture inductance  cannot  be  economically  reduced  below  a 
value  sufficient  to  prevent  a  destructive  flow  of  current, 
as  was  shown  in  the  example  in  Section  94. 

As  showing  the  gain  in  stability  of  operation  by  excit- 
ing the  motor  somewhat  above  that  which  would  result 
in  the  minimum  current,  were  there  no  armature  reac- 
tions present,  Mr.  Kapp  gives  the  following  theoretical 
table.* 

TABLE   SHOWING   WORKING   CONDITION   OF 
.    TRANSMISSION   PLANT. 

Total  resistance  in  circuit,  I  ohm;   total  reactance,  4  ohms. 


Generator  excited  to  give      ....     .... 
Motor  excited  to  give  *    '* 

1  100 

1  200 

IIOO 
I^OO 

I  IOO  VOltS 

i  -7  co  volts 

Normal  power  given  off  by  motor  . 
Maximum  power  given  off  by  motor  be- 
fore breaking  from  synchronism 
Margin  of  excess  load  causing  breakdown 
of  the  system 

125 
200 
60 

125 
250 

IOO 

125  H.P. 
268  H.P. 

With  a  smaller  impedance  in  circuit,  the  possible 
overload  before  the  motor  breaks  from  synchronism 
would  be  greater,  as  is  shown  by  the  considerations  just 

*  Electrical  Transmission  of  Rnergy,  4th  ed.,  p.  277. 


586 


ALTERNATING   CURRENTS. 


discussed,  the  experiments  of  Kolben,*  and  the  experi- 
ence in  American  plants. 

157.  Experiments  of  Bedell  and  Ryan.  —  Bedell  and 
Ryanf  made  a  series  of  experiments  on  a  pair  of  di- 
minutive eight-pole,  smooth-core  Westinghouse  alterna- 
tors, giving  a  frequency  of  139,  one  of  which  was  run 
as  a  motor,  and  the  other  as  a  generator.  (The  machines 
were  built  to  each  supply  ten  16  C.P.  lamps.)  The  re- 
sistance of  the  machine  circuit  was  .31  ohm,  and  the 
self-inductance  of  the  motor  armature  .32  millihenry. 


m 

Ul 

K 

§30 
o 

g 
I85 

20 

\ 

% 

V 

± 

V 

& 

1234 
MOTOR   FIELD  CURRENT 

Figr.  271 


The  curve  of  magnetization  of  the  motor  was  practically 
a  straight  line.  It  was  found  that  the  motor  would  oper- 
ate only  under  field  excitations  varying  from  1.5  to  3.5 
amperes,  and  required  an  abnormal  armature  current  to 
carry  its  load,  the  minimum  current  being  at  an  excita- 
tion of  3  amperes  (Fig.  271);  also,  that  a  very  small 


*  Elektrotechnische  Zeitschrift,  Vol.  16,  p.  802;  London  Electrical 
Engineer,  1895. 

f  Action  of  a  Single-Phase  Synchronous  Motor,  Jour.  Franklin  Inst., 
March,  1895. 


ALTERNATING-CURRENT   MOTORS. 

increase  of  load  would  throw  it  out  of  synchronism. 
The  load  consisted  of  the  friction  of  a  |  horse-power 
Edison  dynamo.  A  variable  inductance  consisting  of 
a  coil  with  a  movable  iron  core  was  then  inserted  in 
the  circuit.  By  moving  the  core  of  this  coil  it  was  found 
that  when  its  inductance  was  1.68  millihenrys  the  motor 
required  a  minimum  armature  current  for  a  given  load, 
ran  with  stability  through  a  wide  range  of  load,  and 


23*56 
MOTOR   FIELD   CURRENT 

Fig.    272 

operated  at  excitations  of  from  1.8  to  6  amperes.  The 
excitation  was  not  carried  over  6  amperes  as  there  was 
danger  of  springing  the  motor  shaft,  which  was  weak. 
Curve  C,  in  Fig.  272,  shows  the  armature  current  for 
different  excitations  when  the  motor  was  under  the 
same  small  constant  load,  as  in  the  trial  without  ex- 
ternal inductance.  Curves  D,  F,  and  G  show  plainly 
the  tendency  of  the  armature  reaction  referred  to  above 


588  ALTERNATING   CURRENTS. 

to  hold  the  generator  and  motor  excitations  at  the  point 
of  maximum  efficiency.  Curve  D  represents  the  gene- 
rator pressure,  and,  although  the  excitation  was  con- 
stant, the  generator  pressure  rises  as  the  motor  pressure 
is  increased.  This  is  due  to  the  reaction  caused  by  the 
current  swinging  from  a  lag  to  a  lead  with  reference  to 
the  generator  pressure.  At  the  same  time  the  motor 
pressure,  which  is  represented  by  curve  Fy  at  first  is 
larger  and  then  grows  smaller  than  would  be  the  case 
were  no  reactions  present.  The  curve  G  represents  the 
pressure,  considering  reactions  absent.  The  effect  in 
the  motor  is  caused,  as  in  the  generator,  by  the  current 
increasing  its. lead  with  reference  to  the  motor  pressure. 
Figure  273  shows  the  polar  diagrams  for  various  excita- 
tions at  which  the  motor  was  run  under  constant  load. 
OElt  OEft  and  OR  are  the  generator,  motor,  and  result- 
ant pressures  respectively,  and  OC  the  current.  The 
angle  by  which  the  current  lags  behind  the  resultant 
pressure  was  obtained  from  the  impedance  of  the  arma- 
ture circuit.  It  may  be  clearly  seen  from  the  diagrams 
that  the  current  swings  from  a  position  of  large  lag, 
with  reference  to  the  generator  pressure,  at  a  small 
excitation  of  the  motor,  into  phase  with  it,  —  the  point 
of  minimum  current  for  the  given  load  on  the  motor,  — 
and  finally  into  a  position  of  large  lead  when  the  motor 
is  greatly  over-excited. 

This  series  of  experiments  and  the  preceding  discus- 
sions (Sect.  1 56)  show  that  there  was  some  foundation 
for  the  statement  of  earlier  experimenters,  that  alterna- 
tors must  have  self -inductance  in  their  armature  circuits 
if  they  are  designed  to  be  run  in  parallel.  The  appli- 


ALTERNATING-CURRENT   MOTORS.  589 


No.  7 


590         ALTERNATING  CURRENTS. 

cation  of  that  statement  to  the  case,  however,  is  fal- 
lacious, since  alternators  operating  in  parallel  should 
require  much  less  than  the  torque  of  normal  load  to 
hold  them  in  step,  so  that  the  synchronizing  tendency 
of  armatures  with  small  inductance  is  ample  to  make 
them  run  in  parallel,  and  for  either  parallel  working  or 
for  operation  as  synchronous  motors,  a  small  armature 
impedance  is  of  the  greatest  importance. 

158.  Effect  of  Wattless  Current  on  Torque.  —  Since  a 
synchronous  motor  seldom  operates  at  the  exact  load 
for  which  its  excitation  is  adjusted,  the  armature  current 
is  likely  to  have  a  large  wattless  component.  Hence, 
during  a  portion  of  each  half  period  the  motor  armature 
must  return  to  the  circuit  some  of  the  energy  which  was 
delivered  to  it  during  the  remainder  of  the  half  period. 
This  causes  the  torque  of  a  single-phase  armature  to 
vary  from  a  large  positive  value  to  a  small  negative 
value  in  each  half  period,  and  in  order  that  this  effort 
to  return  the  energy  represented  by  the  wattless  current 
may  not  break  it  from  synchronism,  it  is  well  for  the 
armature  to  be  very  solidly  built,  or  to  have  a  fly-wheel 
attached  to  its  shaft.  Since  the  torque  of  a  polyphase 
armature  is  uniform  throughout  the  period  (Sect.  144), 
polyphase  synchronous  motors  are  likely  to  run  more 
satisfactorily  than  single-phasers. 

The  magnitude  of  the  wattless  component  depends 
directly  upon  the  armature  self-inductance  and  the 
amount  of  excitation  given  the  motor.  When  the  arma- 
ture self-inductance  is  small,  the  armature  current  does 
not  differ  greatly  with  different  excitations,  and  hence 
the  wattless  current  in  average  operation  is  reduced. 


ALTERNATING-CURRENT   MOTORS.  591 

This  is  an  additional  advantage  of  motors  having  arma- 
tures with  a  minimum  impedance. 

159.  Rotary-Field  Induction  Motor.  —  The  well-known 
principles  which  cause  the  rotation  of  a  disc  of  copper 
pivoted  above  a  rotating  horseshoe  magnet  have  been 
put  into  use  through  the  discoveries  of  Ferraris,  Tesla, 
Haselwander,    Dobrowolsky,    and   many   others.      The 
arrangements  proposed   by  Tesla  were    doubtless   the 
first  direct  applications  of  these  principles  to  commer- 
cial use,  in  which  they  are  destined  to  play  a  large  part 
in    the  transmission   and  distribution    of   power.*     An 
almost  simultaneous  publication  of  a  series  of  scientific 
experiments  by  Ferraris  shows  the  operation  of  similar 
apparatus,  f  and  various  experiments  of  a  similar  nature 
or  for  a  similar  purpose  are  on  record.     Each  of  these 
experiments  caused  an  iron  or  copper  armature  to  rotate 
when  placed  within  the  region  of  a  rotating  magnetic 
field. 

160.  A  Rotating   Magnetic   Field.  —  If  two  coils   of 
wire  are  arranged  at  right  angles  so  as  to   enclose  a 
cylindrical  iron  core,  or  if  two  pairs  of  coils  are  placed 
at  right  angles  on  a  ring  core  (Fig.  250),  the  magnetism 
set  up  in  the  core  when  a  current  is  passed  through  the 
coils  is  the  resultant  of   the  magnetization  due  to  the 
two  coils. 

If  the  magnetizing  currents  are  two  sinusoidal  alter- 
nating currents  with  90°  difference  of  phase,  then,  at 

*  A  New  System  of  Alternate-Current  Motors  and  Transformers, 
Trans.  Amer.  Inst.  E.  £.,  Vol.  5,  p.  308. 

t  Electro-dynamic  Rotation  by  Means  of  Alternating  Currents,  Lon- 
don Electrician,  Vol.  21,  p.  86. 


592         ALTERNATING  CURRENTS. 

any  instant,  the  magnetizing  force  due  to  one  of  the 
coils  is  //!  =  Hm  sin  a,  and  of  the  other  coil  is 

ffz  ==  Hm  sin  (a  —  90°)  =  Hm  cos  a, 

where  Hm  is  the  maximum  magnetizing  force  of  either 
coil  (Fig.  274).  The  resultant  magnetizing  force  is  then 
HR  =  V//J2  +  //22  =  Hmt  and  is  therefore  constant  in 
magnitude.  The  direction  of  this  constant  magnetizing 
force  is  variable.  When  a  =  o°,  HR  lies  in  the  plane  of 
the  first  coil,  and  when  a  —  90°,  HR  lies  in  the  plane 
of  the  other  coil.  The  magnetizing  force  of  each  coil 
has  a  sinusoidal  or  harmonic  variation,  and  the  result- 
ant magnetizing  force  is  the  resultant  of  two  harmonic 
variations  with  90°  difference  of  phase.  As  is  well 
known,  such  a  resultant  has  a  uniform  magnitude  and 
a  uniformly  varying  direction.  The  instantaneous  values 
of  the  resultant  may  therefore  be  diagrammatically  rep- 
resented by  the  instantaneous  positions  of  a  line  of  fixed 
length,  rotating  at  a  uniform  rate  around  one  end,  such 
as  OHB  in  Fig.  274. 

If  the  maximum  ampere-turns  of  one  coil  are  greater 
than  those  of  the  other  coil,  the  magnitude  of  the  result- 
ant magnetizing  force  varies.  The  rotating  field,  in 
this  case,  may  be  diagrammatically  represented  by  a 
uniformly  rotating  line,  which  varies  in  length,  so  that 
its  tip  traces  an  ellipse  whose  minor  and  major  axes  are 
respectively  in  the  planes  of  the  stronger  and  weaker 
coils.  If  the  windings  of  the  coils  are  similar,  and, the 
currents  equal,  but  the  phase  difference  is  not  90°,  a 
variable  field  again  results. 

If  the  phases  of  the  two  currents  are  in  unison, 

*=  A/2  Hm  sin  a. 


ALTERNATING-CURRENT   MOTORS. 


593 


This  shows  that  when  the  two  currents  are  in  unison 
HR  varies  with  sin  a,  and  therefore  varies  from  zero  to 
a  maximum  of  V2  Hm,  but  its  direction  must  be  constant, 
since  the  values  of  its  two  components  are  equal  at  every 
instant.  Its  direction  evidently  lies  in  a  plane  between 
the  planes  of  the  two  coils.  The  diagrammatic  represen- 
tation of  the  resultant,  here,  is  a  line  of  fixed  direction 


Fig.  274 

which  harmonically  varies  in  length,  the  total  range  of 
variation  being  from  —  A/2  Hm  to  -f  A/2  Hm. 

For  any  difference  of  the  current  phases  between 
zero  and  90°,  both  the  magnitude  and  direction  of  HR 
again  vary,  and  the  diagrammatic  representation  is  again 
a  line  with  its  tip  tracing  an  ellipse.  The  ratio  of  the 
two  axes  depends  upon  the  phase  difference  of  the  cur- 
rents. If  the  currents  have  90°  phase  difference,  but 
the  planes  of  the  coils  are  not  90°  apart,  the  effect  on 
the  resultant  magnetizing  force  is  evidently  the  same 

2Q 


594 


ALTERNATING    CURRENTS. 


as  if  the  conditions  were  reversed.  If  the  currents  are 
not  sinusoidal,  the  value  of  the  resultant  magnetizing 
force,  HRy  varies  in  a  more  or  less  irregular  manner. 
Thus,  Fig.  275  a  indicates  in  a  general  manner  the 
strength  of  the  field  at  different  angular  positions 
when  a  peaked  current  is  applied  to  two  coils  having 
90°  difference  of  position,  and  Fig.  275  b  is  the  same 
for  a  flat-topped  current  curve  having  the  same  maxi- 

A 


Fig.  275  a 

mum  value.  The  dotted  circles  in  each  case  represent 
the  rotating  magnetizing  force  due  to  sinusoidal  cur- 
rents in  the  same  coils. 

The  same  argument  may  be  readily  seen  to  apply 
to  the  resultant  magnetizing  force  due  to  any  number 
of  coils  surrounding  a  core.  When  equal  coils  are  at 
equal  angular  distances,  and  equal  currents  in  the  in- 
dividual coils  differ  in  phase  by  an  amount  equal  to 
the  angular  distance  of  the  coils  from  each  other,  the 
resultant  magnetizing  force  is  always  uniform  in  mag- 


ALTERNATING-CURRENT   MOTORS. 


595 


nitude  and  rotates  at  a  uniform  rate,  provided  the  cur- 
rents are  sinusoidal,  and  its  value  is  HR  =  —  Hm,  where 

m  is  the  number  of  phases*  (compare  Sect.  144).     The 

A 


correctness  of  these  deductions  is  directly  indicatecj 
by  experiment. 

The  Germans  call  the  rotating  field  Drehfelde,  and 
the  polyphase  currents  which  set  up  a  rotating  field 
Drehstrom,  or  rotating  current. 

161.  Action  of  a  Short-circuited  Armature  Winding 
within  a  Rotating  Field.  —  If  a  drum  core  of  laminated 
iron  be  properly  pivoted  within  a  ring,  on  which  coils  are 
so  situated  that  the  field  rotates,  it  will  be  dragged  into 
rotation  by  the  magnetic  pull.  If  the  pivoted  core  be 
of  copper,  it  will  be  dragged  into  rotation  by  the  re- 
actions of  the  foucault  currents  which  are  developed 
in  the  core.  This  is  directly  analogous  to  the  ex- 

*  E.  Arnold,  Elektrotechnische  Zeitschrift,  Vol.  14,  p.  42. 


596 


ALTERNATING   CURRENTS. 


periment  with  the  Arago  disc,  to  which  reference  has 
already  been  made  (Sect.  159). 

In  the  case  of  either  a  solid  core  or  Arago  disc, 
the  foucault  currents  are  not  constrained  in  position, 
and  therefore  take  the  path  of  least  resistance.  The 
result  is  that  much  of  the  effectiveness  of  the  currents 
in  bringing  about  a  rotation  is  lost,  and  the  efficiency 
of  the  device  is  small.  If,  in  the  disc  experiment,  the 
disc  be  cut  up  into  an  indefinitely  large  number  of 
fine  radiating  wires  which  are  connected  together  at 


Fig-.  276 

their  inner  and  outer  ends,  the  useless  or  parasitic 
eddies  may  in  a  large  measure  be  done  away  with, 
and  the  efficiency  of  the  device  be  considerably  raised. 
In  the  same  way  the  drum  core  may  be  made  of 
laminated  iron  in  order  that  the  magnetic  circuit  shall 
be  of  small  reluctance,  and  embedded  in  this  may  be 
copper  wires  which  cross  the  face  of  the  core  and 
are  all  short-circuited  by  copper  rings  at  the  ends 
(Fig.  276).  These  make  constrained  paths  for  the  in- 
duced currents,  and,  if  the  core  is  sufficiently  lami- 
nated and  the  copper  conductors  are  not  too  thick,  the 


ALTERNATING-CURRENT   MOTORS.  597 

parasitic  eddies  are  largely  done  away  with,  and  the 
efficiency  of  such  a  motor  may  be  made  quite  large. 

162.  Variation  in  a  Rotating  Field.  —  There  has  been 
considerable  dispute  regarding  the  uniformity  of  the 
strength  of  the  rotating  field  in  motors  of  this  class. 
The  question  at  issue  being  whether  the  effective  mag- 
netizing force  at  each  instant  is  equal  to  the  sum  of 
the  ampere-turns  on  the  coils,  or  the  ampere-turns  are 
compounded  to  gain  the  resultant  effect  according  to 
the  parallelogram  of  forces.  The  latter  assumption  is 
made  in  the  discussion  given  above  (Sect.  160).  Do- 
browolsky,  Pupin,*  and  others  have  taken  the  other 
view,  and  have  determined  from  that  standpoint  that 
there  is  a  fluctuation  of  about  40  per  cent  in  the 
strength  of  the  field  due  to  two  sinusoidal  currents 
with  90°  difference  of  phase,  and  about  14  per  cent 
fluctuation  in  the  field  due  to  three  .sinusoidal  cur- 
rents with  120°  difference  of  phase. 

With  a  view  of  experimentally  determining  which 
assumption  is  correct,  Messrs.  Hanson  and  Webster 
undertook,  in  the  electrical  laboratories  of  the  Univer- 
sity of  Wisconsin,  the  experimental  measurement  of 
the  strength  of  the  rotating  field  of  a  three-phase  motor, 
when  magnetized  with  three  sine  currents  with  phase 
differences  of  120°.  For  this  purpose  they  placed  a 
test  coil  on  the  surface  of  the  motor  armature  and 
arranged  the  armature  so  that  it  could  be  readily  rotated 
through  a  small  arc  of  fixed  value.  The  reading  of  a 
ballistic  galvanometer  connected  to  the  test  coil  was 
therefore  proportional  to  the  number  of  lines  of  force 

*  Trans.  Amer.  Inst.  E.  £.,  Vol.  8,  p.  562. 


598  ALTERNATING   CURRENTS. 

cut  by  the  coil  when  the  armature  was  rotated.  The 
magnetization  was  effected  by  continuous  currents  in 
the  windings  of  the  motor  fields,  which  were  so  adjusted 
as  to  give  the  proper  phase  relation  to  each  other. 
Thus,  calling  the  coils  a,  b,  and  c,  and  supposing  the 
current  in  a  is  desired  to  be  the  instantaneous  zero 
value  of  the  current,  then  the  current  in  b  must  be 
adjusted  so  that 


and  the  current  in  c  must  be  adjusted  so  that 
Ce  —  cmax  sin  240°. 

The  resultant  magnetism  thus  produced  is  equal  to 
the  instantaneous  magnetization  due  to  an  alternating 
current  taken  at  a  corresponding  instant.  To  get  the 
instantaneous  magnetization  for  any  other  phase  of 
the  alternating  "currents,  the  test  currents  must  be  so 

adjusted  that 

Ca  =  cmax  sin  a, 


The  algebraic  sum  of  the  currents  must  always  be  equal 
to  zero.  The  apparatus  was  arranged  somewhat  as  in 
Fig.  277.  By  the  method  thus  outlined  it  was  found 
that  the  magnetization  due  to  the  field  windings  ad- 
vanced uniformly  as  a  wave  of  fixed  magnitude,  as 
closely  as  the  limits  of  error  of  the  experiment  would 
show.  As  these  errors  were  well  within  2  or  3  per 
cent,  the  experiments  prove  : 

I.    That  the  resultant  magnetizing  force  due  to  the 


ALTERNATING-CURRENT   MOTORS. 


599 


several  coils  arranged  as  in  the  rotary-field  motor  is, 
for  practical  purposes,  equal  to  the  magnetizing  effects 
of  all  the  coils  compounded  according  to  the  ordinary 
methods  of  composition  of  harmonic  variation. 


MOTOR  WINDINGS 


AMPEREMETER 


Fig.  277. 

2.  That  the  magnetization  set  up  is  practically  pro- 
portional to  the  magnetizing  force  when  the  induction 
is  not  pushed  too  high. 

To  determine  to  what  extent  the  saturation  of  the 
iron  in  the  magnetic  circuit  affects  the  latter  deduction, 


600  ALTERNATING   CURRENTS. 

Hanson  and  Webster  made  tests  which  covered  a  con- 
siderable range  of  maximum  currents,  and  which  were 
carried  above  the  bend  in  the  curve  of  magnetization  of 
the  motor.  The  deductions  given  above  appeared  to  be 
practically  correct  within  the  limits  of  the  experiments.* 
Similar  experiments  have  been  proposed  and  carried 
out  by  du  Bois-Reymond,f  Blondel,  J  Behn-Eschenburg, 
and  others. 

163,  Distinction  between  Armature  and  Field.  —  There 
is  some  ambiguity  in  the  designation  of  the  armature 
and  fields  of  induction  motors,  since  it  is  not  uncommon 
to  make  them  with  revolving  field  cores,  and  both  fields 
and  armature  carry  an  alternating  current,  but  the  fol- 
lowing definitions  avoid  all  ambiguities.  The  Field  is  the 
core  upon  which  are  placed  windings  connected  to  the 
external  circuit.  The  current  in  the  fields  is  therefore 
due  to  the  impressed  pressure  of  the  external  circuit. 
The  Armature  is  the  part  of  the  motor  in  the  conductors 
of  which  current  is  induced  by  the  revolving  magnetism 
of  the  fields.  Since  the  armature  current  is  wholly  in- 
duced by  action  of  the  fields,  these  motors  are  called 
Induction  Motors.  With  these  definitions,  it  is  readily 
seen  that  the  induction  motor  acts,  in  many  respects, 
like  a  transformer,  the  primary  winding  of  which  is  on 
the  fields,  and  the  secondary  winding  on  the  armature. 

*  Jackson,  Three-Phase  Rotary  Field,  Electrical  Journal,  Vol.  I,  p.  185. 
Also  see  Pupin,  Trans.  Amer.  Inst.  E.  £.,  Vol.  1 1,  p.  549. 

t  Theoretical  and  Experimental  Study  of  Polyphase  Currents,  Elektro- 
technische  Zeitschrift,  Vol.  12,  p.  303;  Electrical  World,  Vol.  17,  p. 

477- 

\  Elementary  Theory  of  Rotary-Field  Apparatus,  La  Lumiere  Elec- 
trique,  Vol.  50,  p.  358. 


ALTERNATING-CURRENT  MOTORS.  6oi 

The    Germans   call   polyphase  induction  motors  Dreh- 
strom  Motors. 

The  energy  developed  in  the  secondary  circuit  of  the 
induction  motor  is  expended  in  causing  rotation  of  the 
revolving  part  instead  of  causing  heat  and  light  in 
the  external  circuit,  as  is  the  case  of  the  ordinary  trans- 
former. The  same  general  methods  apply,  in  designing 
these  motors,  that  apply  in  designing  transformers. 

164.  Wattless   Magnetizing   Current.  —  Since   an   air 
space  must  be  made  in  the  magnetic  circuit  to  allow 
the  motors  to   operate,  it  is  evident  that  the  wattless 
magnetizing  current  of  induction  motors  must  be  mate- 
rially greater  than  that  of  transformers.      In  fact,  the 
no-load  current  of  some  comparatively  small  motors  of 
this  type,  which  show  quite  a  high  efficiency,  is  entirely 
comparable  to  the  full-load  current.    To  reduce  the  watt- 
less current  to  a  reasonable  limit,  every  effort  must  be 
bent  to  decrease  the  reluctance  of  the  air  space.    As  the 
armature  conductors  may  be  embedded  in  the  armature 
core,  it  is  possible  to  make  the  air  space  simply  that 
required  for  mechanical  clearance,  and,  by  care  in  the 
workmanship,  this  may  be  made  very  small  compared 
with  the  air  space  of  dynamos  built  according  to  the 
ordinary  methods. 

165.  Motor  Speeds  and  Slip.  — The  velocity  of  rotation 
of  the  magnetic  field  depends  upon  the  frequency  of  the 
current  supplied  to  the  motor,  and  the  number  of  pairs 
of  poles  in  the  field.     In  two-pole  machines,  the  number 
of  rotations  which  the  field  makes  per  second,  or  the 
Field  Frequency,  is  equal  to  the  current  frequency,  and, 
in  multipolar  machines,  the  field  frequency  is  equal  to 


602.  ALTERNATING   CURRENTS. 

the  current  frequency  divided  by  the  number  of  pairs  of 

poles,  or  Z. .     The  number  of  pairs  of  poles  which  is  re- 

P 
ferred  to  is  the  number  in  the  rotating  field.  This  is  equal 

to  the  number  of  pairs  of  poles  set  up  by  the  windings 
in  fields  with  a  smooth  magnetic  surface,  but  is  equal  to 

— ?—  times  the  number  of  salient  poles  in  salient-pole  ma- 
2m 

chines  (m  being  the  number  of  phases).  The  latter  can 
scarcely  be  said  to  give  a  uniformly  rotating  field  unless 
there  are  m  crowns  of  poles. 

The  velocity  of  rotation  of  the  armature  can  never 
equal  the  velocity  of  rotation  of  the  field  magnetism, 
since  the  armature  conductors  must  be  cut  by  the  lines 
of  force  of  the  fields  in  order  that  an  electrical  pressure 
may  be  developed  in  the  armature ;  that  is,  the  field 
magnetism  must  always  have  a  relative  velocity  of  rota- 
tion with  reference  to  the  armature  conductors.  In  any 
machine,  the  relative  velocity  is  v  —  V  —  V ,  where  V 
and  V  are  respectively  the  number  of  revolutions  per 
minute  of  the  field  magnetism  and  the  armature  conduct- 
ors. This  relative  velocity  is  called  the  armature  Slip, 
and  is  small,  seldom  exceeding  5  per  cent  of  the  speed 
of  the  motor.  Since  the  current  in  the  armature  must 
be  proportional  to  the  work  done  by  the  motor,  it  must 
vary  with  the  load,  and  v  must  increase  as  the  load  is 
increased.  A  little  consideration  shows  that,  if  the 
magnetism  remains  constant,  the  variation  of  v  with 
the  load  must  be  just  sufficient  to  counterbalance  the 
drop  of  pressure  caused  by  the  current  flowing  in  the 
armature  conductors. 

A  variation  of  v  demands  a  variation  of  V  of  equal 


ALTERNATING-CURRENT   MOTORS.  603 

magnitude,  since  V  is  fixed  by  the  frequency  of  the  cur- 
rent delivered  to  the  motor ;  consequently,  the  speed 
regulation  of  a  rotary-field  motor  is  directly  dependent 
upon  the  loss  of  pressure  in  the  armature  conductors  if 
we  neglect  the  effect  of  armature  reactions  and  drop  of 
pressure  in  the  primary  windings.  This  is  entirely  anal- 
ogous to  the  case  of  continuous-current  shunt-wound 
motors. 

At  starting,  the  relative  velocity  of  the  field  magnetism 
and  the  armature  is  evidently  F,  since  V'  is  zero.  The 
armature  current  is  therefore  very  great,  and  the  starting 
torque  may  also  be  very  great  provided  the  armature 
reactions  do  not  too  greatly  disturb  the  field.  To  avoid 
injury  to  the  armature  from  the  current  at  starting, 
means  must  be  taken  to  prevent  its  becoming  excessive, 
exactly  as  in  the  case  of  continuous-current  machines 
worked  on  constant  pressure. 

166.  Graphical  Illustration  of  Relations  in  Induction 
Motors. — The  reactions  of  the  polyphase  induction 
motor  may  be  set  forth  very  clearly  by  graphical  repre- 
sentation. Suppose  we  have  under  consideration  a  two- 
phase  motor,  as  shown  diagrammatically  in  Fig.  278, 
where  aa'  and  bb'  are  two  pairs  of  coils  in  series,  each 
pair  being  connected  to  a  pair  of  two-phase  feeders. 
The  armature  we  will  suppose  for  convenience  is  of  the 
squirrel-cage  or  short-circuited  bar  type.  That  is,  the 
conductors  are  embedded  in  the  face  of  the  armature 
and  are  short-circuited  by  rings  extending  around  the 
armature  at  each  end  (see  Fig.  276).  From  the  fore- 
going discussion  (Sect.  160)  it  is  evident  that  a  mag- 
netic north  pole  on  one  side  and  a  magnetic  south  pole 


604 


ALTERNATING   CURRENTS. 


just  opposite,  will  rotate  around  the  field  core  with  the 
frequency  of  the  alternating  current  (/).  This  mag- 
netic field  will  induce  under 
it,  in  the  conductors  of  the 
armature  which  it  cuts,  a 
pressure  which  causes  a  cur- 
rent to  flow  in  the  conduc- 
tors. This  current,  if  the 
armature  is  free  from  self- 
inductance,  will  set  up  a 
magnetic  field  lagging  90° 
Fig.  278  behind  the  magnetism  set 

up  by  the  field  windings  (Fig.  278  #).  Therefore  let  OB 
in  Fig.  279  be  the  strength  of  the  rotary  field  which  pro- 
duces in  the  armature  the  resultant  current,  OCa.  Then 
OA  may  be  represented  as  the  field  due  to  this  armature 


Fig.  278  a 

current.  The  impressed  magnetizing  force  must  be 
sufficient  to  supply  the  field  OB  and  overcome  OA,  or 
must  be  sufficient  to  set  up  a  field  OM.  OCf  repre- 
sents the  relative  phase  and  magnitude  of  the  field  cur- 


ALTERNATING-CURRENT   MOTORS. 


605 


rent,  provided  the  number  of  primary  conductors  is 
equal  to  the  number  of  secondary  conductors.  This 
construction  requires  us  to  consider  the  polyphase 
currents  combined  at  every  instant  into  a  resultant 


A 
Pig-.  279 


which  may  be  represented  by  the  sum  of  the  vertical 
projections  of  the  polyphase  current  vectors.  This 
assumption  simplifies  the  construction  very  much  and 
enables  the  use  of  exactly  the  same  method  as  that 
used  for  transformers  (Sect.  118);  namely,  in  Fig.  280, 


6o6 


ALTERNATING  CURRENTS. 


if  OC^  is  the  magnetizing  ampere-turns  required  to  set. 
up  the  desired  field,  and  OCa  the  armature  ampere-turns, 
then  the  field  ampere-turns  must  be  OCf.  Also  the 


Pig.  28O 

self-induced  pressure  in  the  fields  will  be  OElt  drawn  to 
the  proper  scale,  while  OE  is  the  pressure  that  must  be 
applied  to  the  fields  when  OA  is  the  element  of  pressure 
which  multiplied  by  the  current  furnishes  the  motor 
losses. 


ALTERNATING-CURRENT   MOTORS.  607 

167.  Torque  of  Ideal  Motor.  —  It  is  evident  that  in  a 
motor  giving  this  diagram,  the  starting  torque  will  be 
enormous  for  a  low-resistance  armature,  as  the  current 
induced  will  be  enormous,  since  the  wattless  magnetizing 
current,  and  hence  the  resultant  magnetic  field,  remains 
practically  constant  if  there  are  no  armature  reactions. 
The  torque  is  of  course  proportional  to  the  field  mul- 
tiplied by  the  current,  or  to  OBxOCa  (Fig.  279).     As 
the  motor  comes  up  to  speed,  the  current  will  decrease 
directly  as  the  speed  increases,  since  the  relative  speed 
or  slip  of  the  armature  with  reference  to  the  rotating 
field  decreases  directly  as  the  speed  increases.     Hence, 
neglecting  armature  reactions  and  self-inductance,    the 
torque  will  be  a  maximum  with  the  armature  standing 
still  and  will  gradually  decrease  to  zero  as  the  armature 
speed  increases  towards  its  limit,  which  is  synchronism 
with  the  rotating  field. 

168.  Effect  of  Magnetic  Leakage.  — As  there  must  be 
clearance  between  the  armature   and  field,   a  path  is 
ma^de  for  magnetic  leakage,  which,  on  account  of  the 
opposing  action  of  the  armature  and  field  magnetism, 
becomes  of  very  considerable  magnitude  at  large  loads. 
Lines    of   force   set   up   by   the  armature  and  leaking 
through  this  clearance  space  cause  the  armature  current 
to  lag  exactly  as  though  self-inductance  were  introduced 
into  the  windings,  as  is  also  the  case  in  the  fields.     This 
effect   materially   alters   the   diagram,  as   is   shown   in 
Fig.    281,   where   OE^   and  OEls   are  respectively  the 
armature  and  field  reactive  pressures.     The  diagram  in 
this  case  is  also  drawn  exactly  as  in  the  case  of  trans- 
formers, where  there  is  self-inductance  in  the  primary 


6oS 


ALTERNATING   CURRENTS. 


and  secondary  coils,  and  gives  an  impressed  field  press- 
ure OEf  and  a  field  current  OCf  lagging  by  the  angle 
<$>f.  The  angle  of  lag  </>a  in  the  armature  evidently  may 


Fig.  281 

cause  a  serious  decrease  in  the  motor  torque  from  two 
causes;  first  by  decreasing  the  armature  current  for  a 
given  induced  pressure,  and  second  by  retarding  the  phase 


ALTERNATING-CURRENT   MOTORS. 


609 


of  the  current  with  respect  to  the  magnetic  field.  For  this 
reason,  added  to  its  effect  on  the  slip,  induction  motors 
are  built  to  give  as  small  a  leakage  field  as  possible. 
Figure  282  indicates  a  method  of  winding  frequently 
employed,  which  makes  it  possible  to  reduce  the  leakage 
field  to  a  minimum  by  reducing  the  air  space.  The 
windings  are  placed  in  evenly  distributed  slots,  thus 


Pig.  282 

avoiding  polar  projections.  As  the  frequency  of  alter- 
nation in  an  armature  bar  is  a  maximum  when  the 
motor  is  at  rest,  its  reactance  is  then  a  maximum,  and 
the  armature  current  is  caused  to  have  a  maximum  lag 
with  respect  to  the  induced  pressure  ;  and  the  torque 
for  a  given  current  is  reduced  in  proportion  with  the 
cosine  of  the  lag,  cos  <t>a.  Since 


cos  $.  =     -, 


6lO         ALTERNATING  CURRENTS. 

it  is  evident  that  by  increasing  the  resistance  of  the 
armature  conductors  at  starting,  cos  $a  may  be  increased, 
and  the  starting  torque,  which  is  equal  to 

CaE  cos  <t>a  mv*,  cos  <f>a  * 

m  —     — 777 —  =  J\. ^ , 

27rl/'  Impedance 

may  be  increased  to  a  maximum. 

The  constant,  K,  in  this  formula  depends  upon  the 
winding  and  dimensions  of  the  armature  and  the 
strength  of  the  magnetic  field.  In  practice  an  external 
resistance  is  usually  introduced  by  some  mechanical 
device  into  the  armature  windings,  at  starting,  which 
serves  both  to  increase  the  torque  at  starting  and  to 
avoid  the  excessive  rush  of  current  which  might  occur 
with  the  armature  stationary.  Figure  283  shows  the  rela- 
tion of  torque  to  slip  for  an  armature  having  a  reactance 
of  .18  and  resistances  of  .02,  .045,  .18,  and  .75  ohms.f 
This  shows  plainly  that  the  torque  can  be  caused  to 
have  a  maximum  value  up  to  a  slip  equal  to  10  per 
cent  of  the  field  frequency  by  gradually  reducing  the 
resistance  of  the  armature  circuit  from  .18  to  .02  ohms 
as  the  speed  of  the  armature  increases.  The  relations 
are  as  follows : 

*  Ca=-  -=-.k—,  where  k  is  a  constant  depending  on  the  strength  of 

w 
the  field  and  the  number  of  armature  conductors  ;   E  =  kV;   K  =  — . 

27T 

E  is  the  pressure  that  would  be  induced  in  the  armature  windings  if  the 
armature  were  rotated  at  a  speed  of  V  revolutions  per  minute  in  a  sta- 
tionary field  equal  in  magnitude  to  the  rotating  field;  e  is  the  pressure 
that  wpuld  be  induced  under  similar  conditions  at  a  speed  v;  v\  is  the 
frequency  with  which  the  field  cuts  the  armature  conductors  when  the 

slip  is  z/,  and,  therefore,  v\  =2-—.     m  is  the  number  of  phases. 
60 

t  Steinmetz,  Trans.  Amer.  Inst.  E.  E.,  Vol.  XI.,  p.  760. 


ALTERNATING-CURRENT   MOTORS. 


/«      /.5 

and  the  torque  is  therefore  equal  to 
mK-^- 


/ 


STANDSTILL 


\ 


Fig.  283 

which  is  a  maximum  when  27rv-^La  =  Ra.  When  the  arma- 
ture is  at  rest,  v:  =f  and  to  give  maximum  torque  Ra 
must  be  equal  to  2  TrfLa.  If  Ra  is  either  greater  or  less, 


6l2         ALTERNATING  CURRENTS. 

the  torque  is  reduced,  for  armature  at  rest.  If  Ra  is 
greater  than  2  7rfLa,  the  torque  continuously  decreases 
as  the  speed  of  the  armature  rises ;  while  if  Ra  is  less 
than  2  7r/Za,  the  torque  reaches  a  maximum  when  v  has 
such  a  value  that  Ra=  2  Trv-^L^  If  Ra  is  very  small 
compared  to  2  7r/Za,  the  starting  torque  is  very  small, 
and  the  torque  increases  to  a  maximum  which  occurs 
at  a  slip  very  near  to  synchronism. 

Induction  motors  are  usually  designed  to  run  at  a 
speed  which  is  between  synchronism  and  the  speed 
giving  the  greatest  torque.  In  designing  them,  La  is 
made  the  least  possible,  and  Ra  is  then  given  such  a 
value  that  the  slip  at  normal  full  load  is  sufficient  to 
give  a  value  of  the  torque,  which  is  from  one-half  to 
three-fourths  of  its  maximum  value.  Such  motors  can 
therefore  carry  considerable  overloads,  but  if  the  re- 
sisting moment  of  the  load  is  increased  beyond  the 
maximum  torque,  the  motor  stops.  In  this  respect,  in- 
duction motors  differ  from  continuous-current  motors 
operated  on  a  constant  pressure,  in  which  the  torque 
increases  in  direct  proportion  with  the  armature  current 
and  therefore  with  the  resisting  moment  of  the  load, 
provided  the  total  magnetism  passing  through  the  ar- 
mature remains  constant.  Increasing  the  load  on  a 
continuous-current  motor  will  not  stop  it  until  the  arma- 
ture burns  up  or  the  drop  of  pressure  due  to  current 
flowing  through  the  armature  conductors  is  equal  to  the 
impressed  pressure. 

169.  Forms  of  Armature  Windings.  —  The  armature 
windings  of  induction  motors  may  be  of  either  drum  or 
ring  types,  though  the  drum  type  is  most  commonly 


ALTERNATING-CURRENT   MOTORS.  613 

used.      The  arrangement  of  the  windings    may  be    of 
three  forms  : 

1.  Squirrel-cage  form,  in  which  single  embedded  bar 
conductors  are  placed  on  the  armature  core  and  all  con- 
nected together  at  each   end  by  a  copper  ring,  thus 
making  a  conductor  system  similar  in  form  to  the  re- 
volving cylinder  of  a  squirrel   cage  (Fig.    276).      The 
conductors  are  insulated  from  the  core. 

2.  Independent  short-circuited  coils.     In  this  form  of 
winding  the  armature  conductors  are  of  insulated  wire 
wound   in   independent    short-circuited  coils,  or  of  in- 
sulated bars  connected  by  end  connectors  in  such  a  way 
as  to  make  independent  short-circuited  coils. 

3.  Independent  coils  short-circuited  in  common.     Here 
the  coils  are  wound  as  in  the  preceding  form,  but  in- 
stead of  being  short-circuited  independently,  all  the  ends 
are  brought  to  a  common  point,  or  pair  of  points,  one 
of  which  may  be  at  the  front  end  and  the  other  at  the 
back  end  of  the  armature. 

It  is  evident  that  the  pitch  of  the  coils  of  the  second 
and  third  forms  of  drum  windings  must  be  equal  to  an 
odd  number  of  times  the  pitch  of  the  field  poles,  in  order 
that  the  electrical  pressure  set  up  in  the  conductors 
may  be  additive,  and  coils  may  therefore  be  diametral 
or  chordal  in  machines  with  an  odd  number  of  pairs  of 
poles,  but  cannot  be  diametral  in  machines  with  an 
even  number  of  pairs  of  poles.  The  actual  number  of 
coils  is  a  matter  of  perfect  freedom,  provided,  it  is  a 
multiple  of  two  or  three  and  the  connections  of  con- 
ductors be  properly  made  so  that  the  armature  surface 
may  be  uniformly  covered.  A  three-coil  winding  for  a 


614 


ALTERNATING  CURRENTS. 


drum  armature,  which  is  intended  to  surround  the  re- 
volving field  of  an  eight-pole  machine,  is  shown  dia- 
grammatically  in  Fig.  284,  and  a  three-coil  armature 


Fig.  284 


Fig.  285 


which  is  intended  to  revolve  within  a  six-pole  field,  is 
shown  in  Fig.  285. 

170.  Field  Windings. —  The  field  windings  of  induction 
motors  are  almost  always  arranged  to  produce  more  than 
two  poles,  in  order  to  bring  the  machine  to  a  reasonable 
speed.  The  field  frequency  in  revolutions  per  second, 

as  already  shown  (Sect.  165),  is  equal  to.^-,  where /is  the 

P 
frequency  of  the  alternating  current  and  /  the  number 

of  pairs  of  poles,  and  in  revolutions  per  minute  this  be- 

60  f 
comes  V—  — ~,  and  we  therefore    have  the  following 

table  of  motor  speeds  for  the  frequencies  common  in 
this  country. 

This,  table  shows  the  futility  of  attempting  to  build 
satisfactory  induction  motors  of  small  size,  intended  for 
use  on  even  the  lowest  frequencies  commonly  used  in 
this  country,  with  less  than  six  poles ;  and  on  the  higher 


ALTERNATING-CURRENT   MOTORS. 


6I5 


Frequencies  in  common  use. 

Number  of  poles 

y,  when 

of  motor. 

V,  when 
/=6o. 

V,  when 
/=66§. 

V,  when 
/=«5. 

V,  when 
/-*»• 

y=2s. 

2 

3600 

4OOO 

7500 

8000 

1500 

4 

I800 

2OOO 

375° 

4OOO 

750 

6 

I2OO 

1333 

2500 

2666 

500 

8 

900 

1000 

1875 

2OOO 

375 

10 

72O 

800 

1500 

1600 

300 

12 

600 

666 

1250 

1333 

250 

16 

45° 

500 

937 

IOOO 

187 

20 

360 

400 

75° 

800 

'5° 

24 

300 

333 

625 

666 

I25 

frequencies  not  less  than  twelve  poles  are  required  to 
give  a  reasonable  speed.  Motors  of  greater  output  than 
ten  horse  power  should  have  a  sufficiently  large  number 


Pig.  286 

of  poles  to  give  a  field  velocity  which  does  not  exceed  750 
or  800  revolutions  per  minute.  The  windings  for  this 
purpose  may  be  either  placed  directly  upon  polar  projec- 
tions of  the  field  frame,  as  in  Fig.  286,  which  shows  a 


6i6 


ALTERNATING   CURRENTS. 


four-pole  two-phase  machine,  or  they  may  be  arranged 
as  embedded  conductors  in  a  frame  of  uniform  magnetic 
surface,  as  in  Fig.  287,  which  shows  a  four-pole  three- 
phase  machine.  The  embedded  conductors  may  be 


Fig.  287 

wound  as  either  a  drum  (Fig.  287),  or  ring  (Fig.  287  #) 
field,  and  they  give  the  most  approved  arrangement, 
since  embedding  serves  to  reduce  the  reluctance  of  the 
magnetic  circuit  and  therefore  to  increase  the  power 
factor  of  the  motor.  Since  the  actual  magnet  poles  are 


ALTERNATING-CURRENT   MOTORS.  617 


ccr' 


Fig-.  287  a 


6i8 


ALTERNATING   CURRENTS. 


produced  by  the  resultant  effects  of  the  polyphase  cur- 
rents, it  requires  two-coil  sections  to  produce  each  magnet 
pole  in  a  two-phase  machine,  and  three-coil  sections  in  a 
three-phase  machine.  The  connections  of  the  field  coils 
may  be  traced  out  according  to  the  instructions  given  for 
connecting  the  armature  coils  of  polyphase  generators 
(Sect.  102  a).  The  connections  for  a  three-phase  field  are 
illustrated  by  Fig.  287*2,  which  shows  a  star-connected, 
four-pole  ring  field.  The  star  connection  is  ordinarily 
preferred  for  three-phase  fields,  as  less  pressure  is  im- 


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.     |      ^ 

v 

j  < 

•'^ 

•X* 

' 

-' 

^/* 

^^ 

' 

i^ 

^^ 

-• 

i 

A' 

B 

B 

c 

|C' 

A 

Fig.  287  b 

pressed  on  a  coil,  so  that  fewer  turns  of  wire  are  required 
and  the  strain  on  the  insulation  of  each  coil  is  less.  The 
total  weight  of  copper  is  equal  in  star  and  mesh  connec- 
tions. Figure  287  b  shows  a  development  of  the  eight- 
pole,  star-connected  drum  winding  of  Fig.  288. 

The  frequency  of  the  magnetic  cycles  in  the  iron  of 
the  fields  is  equal  to  the  frequency  of  the  current  flow- 
ing in  the  magnetizing  coils,  but  in  the  armature  it  is 
equal  to  the  motor  slip ;  and  the  hysteresis  and  foucault 
current  losses  per  pound  of  iron  are  therefore  many 
times  greater  in  the  fields.  In  this  respect  the  charac- 


ALTERNATING-CURRENT   MOTORS. 


619 


Fig.  288 


teristics  of  an  induction  motor  are  exactly  the  reverse 
of  those  of  a  continuous-current 
machine;  where  the  field  loss 
consists  of  the  C^R  loss  only, 
while  the  armature  loss  is  the 
sum  of  the  C2R  and  core  losses, 
of  which  the  latter  may  be 
the  larger  portion.  In  the  in- 
duction motor,  the  field-core 
losses  are  large  and  the  arma- 
ture-core losses  are  scarcely 
appreciable  in  a  well-designed 

machine ;  it  is  therefore  desirable  to  reduce  the  amount 
of  iron  in  the  fields  to  the  least  volume  possible,  if  it 
can  be  done  without  increasing  the  magnetic  density. 
For  this  reason,  in  machines  of  considerable  size,  it  is 
usual  to  arrange  the  armature  so  that  it  surrounds  the 
fields,  as  in  Fig.  288,  in  which  case  the  latter  revolves 
and  the  armature  is  stationary.  This  makes  the  use  of 
collector  rings  necessary,  but  their  disadvantages  are 
usually  inconsiderable.  In  all  cases  where  squirrel-cage 
armatures  are  used,  the  number  of  field  conductors 
should  be  an  uneven  multiple  of  the  number  of  arma- 
ture conductors,  in  order  that  there  may  be  no  dead 
points  in  starting. 

171.  Starting  and  Regulating  Devices.  —  Four  differ- 
ent arrangements  may  be  used  for  starting  polyphase 
induction  motors. 

I.  Small  machines  are  commonly  connected  directly 
to  the  circuit  without  the  intervention  of  any  special 
starting  devices.  This  is  not  a  safe  proceeding  for 


620         ALTERNATING  CURRENTS. 

large  machines,  as  when  the  armature  is  at  rest  and 
the  fields  are  directly  connected  to  the  supply  circuit,  the 
machine  is  in  the  condition  of  a  transformer  with  the 
secondary  short-circuited,  and  is  liable  to  burn  up  be- 
fore getting  under  way.  The  Allgemeine  Elektricitats 
Gesellschaft  of  Berlin  arrange  their  smaller  motor  arma- 
tures with  two  rows  of  conductors,  making  two  indepen- 
dent squirrel-cages  (Fig.  289),  one  considerably  farther 


Fig-.  289 

from  the  armature  surface  than  the  other,  which  is 
reported  to  reduce  the  starting  current  of  the  machines. 
2.  (a)  Resistances  in  Field.  Resistances  may  be  in- 
serted in  the  circuits  leading  to  the  motor  fields,  to  be 
used  in  much  the  same  manner  as  starting  resistances 
are  used  in  starting  continuous-current,  constant-pressure 
motors.  Resistances  arranged  in  this  way  in  each  cir- 


ALTERNATING-CURRENT  MOTORS.  621 

cuit  must  be  manipulated  simultaneously,  and  therefore 
must  be  mechanically  coupled.  Starting  rheostats 
similar  to  continuous-current  motor  starting  boxes,  and 
liquid  resistances  arranged  to  be  varied  by  dipping 
plates  in  a  bath,  have  been  used.  Three  rheostats  are 
required  for  a  three-phase  motor,  and  two  for  a  two- 
phase  motor  operated  on  independent  circuits.  Two- 
phase  motors  on  three-wire  circuits  may  be  started  with  a 
single  resistance  inserted  in  the  common  wire,  or  by  two 
resistances  inserted  respectively  in  the  independent  wires. 
The  insertion  of  resistance  in  the  field  circuits  of  in- 
duction motors  serves  to  reduce  the  starting  current  on 
light  loads  by  reducing  the  pressure  at  the  field  ter- 
minals ;  this  also  causes  a  reduction  of  the  magnetism, 
which  is  equivalent  to  reducing  K  in  the  expression  for 

the    starting  torque  (mK—-^ — ^  2a         A  which  shows 
\        -*va    i  <4  ^  ^i  •"»  / 

that  the  starting  torque  under  these  conditions  is  mate- 
rially smaller  than  the  maximum  torque  of  the  armature, 
since  4  7r2v-fLa2  is  bound  to  be  considerably  larger  than 
R?  if  the  armature  CZR  losses  are  not  excessive;  and 
the  motor  must  therefore  take  an  excessive  current  to 
start  a  heavy  load.  This  plan  has  been  used  quite  ex- 
tensively by  European  manufacturers,  especially  for  large 
machines  which  may  be  started  with  belt  on  loose  pulley, 
(b)  Variable  Compensator  or  "  Auto-transformer." 
The  pressure  at  the  terminals  of  the  fields  may  be 
reduced  at  starting  by  introducing  an  impedance  coil 
across  the  supply  circuits  and  feeding  the  motor  from 
variable  points  on  its  windings.  This  arrangement  may 
be  caused  to  supply  a  large  starting  current  without  inter- 


622         ALTERNATING  CURRENTS. 

fering  with  the  supply  circuits,  but  it  has  the  same  effect 
on  the  motor  torque  as  a  resistance  in  series  with  the  fields. 
3.  Resistance  in  Armature.  The  torque  of  the  armature 
at  starting  may  be  made  equal  to  the  maximum  running 
torque  by  inserting  resistances  in  the  armature  circuits 
which  increase  the  total  armature  resistance  at  starting 

rj         .        T-)  jr>  r 

in  the  ratio      e  "*"     a  =  ^-  =  ^-,  where  Re  is  the  exter- 

^-a  ^-a         ^m 

nal  or  starting  resistance  and the  slip  at  full  maxi- 

/                                 R 
mum  torque.    As  already  shown  (Sect.  168),  vn  = — > 

2  7T"_L/£ 

and  therefore  to  get  a  maximum  torque  when  vm  =/, 

requires  that  Ra  +  RC  =/,  or  R.  =  2  7rfLa-Ra,  or  the  ex- 
2  TrLa 

ternal  resistance  in  the  armature  circuit  required  at 
starting  in  order  to  give  maximum  torque  must  equal 
the  difference  between  the  reactance  and  the  resistance 
of  the  armature.  The  total  resistance  used  should  be 
larger  and  arrangements  made  to  reduce  it  gradually. 
As  the  maximum  torque  is  usually  designed  to  occur,  in 
running  with  natural  armature  circuits,  at  a  slip  between 
one  and  one-third  and  two  times  that  corresponding  to 
the  normal  full-load  torque,  the  armature  and  field  cur- 
rents at  maximum  torque  do  not  exceed  twice  the  full- 
load  currents,  so  that  resistances  inserted  in  the  arma- 
ture circuits  serve  the  double  purpose  of  increasing  the 
starting  torque  and  keeping  the  starting  current  within 
bounds. 

This  plan  has  been  largely  used  by  Siemens  and 
Halske,  the  General  Electric  Company,  the  Stanley 
Electric  Manufacturing  Company,  the  Westinghouse 
Company,  and  others.  The  arrangement  of  the  start- 


ALTERNATING-CURRENT   MOTORS.  623 

ing  rheostat  depends  largely  upon  the  type  of  the 
armature  windings  to  which  it  is  applied.  In  arma- 
tures of  the  squirrel-cage  type  the  conductors  may  be 
tipped  at  one  end  with  tapered  german  silver  strips, 
which  come  in  contact  with  a  sliding  copper  ring.  At 
starting,  this  ring  may  just  touch  the  german  silver 
tips,  and  as  the  machine  speeds  up  the  ring  may  be 
slid  along  until  the  tips  are  cut  out  and  the  copper 
armature  conductors  are  directly  connected  together 
through  the  ring. 

If  the  armature  revolves,  it  is  evident  that  the  ring 
must  be  arranged  to  slide  on  a  spline  on  the  shaft,  and 
to  be  controlled  by  a  grooved  sliding  collar  and  loose 
lever.  The  same  device  may  be  used  for  armatures 
with  coils  having  a  common  short-circuiting  point.  In 
this  case  one  set  of  coil  terminals  are  permanently  con- 
nected together,  and  the  other  set  are  connected  into 
german  silver  strips,  which  may  be  short-circuited  by  a 
sliding  ring,  as  already  explained.  This  arrangement 
has  been  used  by  Siemens  and  Halske,  the  General 
Electric  Company,  and  others. 

If  the  armature  windings  are  arranged  so  as  to  have 
but  one  coil  for  each  phase,  the  introduction  of  resist- 
ance is  very  simple,  since  only  one  resistance  coil  for 
each  phase  is  required.  In  this  case,  if  the  armature  is 
stationary,  the  connections  of  the  rheostat  are  made 
directly  into  the  armature  circuits,  or  between  the  ar- 
mature circuits  and  one  point  of  common  connection ; 
while  if  the  armature  revolves,  three  collector  rings 
may  be  placed  on  the  shaft,  and  stationary  rheostats 
may  be  used  to  control  the  resistance  of  the  coils  which 


624  ALTERNATING   CURRENTS. 

are  properly  connected  to  the  rings,  or  the  resistance 
may  be  placed  inside  the  armature  spider  and  controlled 
by  a  sliding  collar  and  loose  lever.  Such  arrangements 
are  used  by  the  Stanley  Electric  Company,  the  General 
Electric  Company,  and  others. 

4.  Commutated  Armature.  The  armature  may  be 
wound  with  the  coils  so  arranged  that  their  conductors 
are  in  series  when  starting  and  in  parallel  when  running. 
If  x  is  the  number  of  parallels  in  such  an  armature,  the 
resistance  at  the  start  is  x*  times  the  running  resistance, 
and  the  reactance  at  start  is  x^  times  the  running  react- 
ance. The  starting  current  in  the  armature  with  the 

conductors  in  series  is  therefore  -  times  as  great  as  it 

x 

would  be  with  the  conductors  in  parallel,  but  the  field 
current  at  starting  is  the  same  with  either  arrangement 
of  the  armature  conductors.  On  the  other  hand,  since 
placing  the  conductors  in  series  increases  the  resistance 
and  reactance  in  the  same  proportion,  the  starting 
torque  of  the  armature  is  the  same  with  the  conduct- 
ors in  series  or  parallel.  The  armature  winding  may 
also  be  so  arranged  that,  instead  of  starting  with  all 
conductors  in  series,  a  portion  of  the  conductors  are 
connected  in  opposition  to  the  others  at  the  start,  and 
are  then  reversed  and  connected  properly  in  series  with 
the  others  after  the  machine  is  in  operation.  The  oppo- 
sition arrangement  affects  the  current  of  both  armature 
and  field. 

The  third  and  fourth  arrangements  may  be  combined 
by  making  the  connectors,  by  means  of  which  the  arma- 
ture conductors  are  placed  in  series,  of  high  resistance 


ALTERNATING-CURRENT   MOTORS.  625 

material  ;  these  connectors  are  cut  out  when  the  con- 
ductors are  short-circuited  together. 

This  device  has  been  used  in  various  forms  by  Sie- 
mens and  Halske  and  the  Westinghouse  Electric  Com- 
pany. 

172.  Effect  of  Rotary  Field  on  Field  Windings.  —  The 
distribution  of  magnetism  in  the  air  space  of  an  induc- 
tion motor  has  been  found  to  be  approximately  sinu- 
soidal, when  the  motor  is  fed  by  sinusoidal  currents,* 
and  the  counter  electric  pressure  set  up  in  the  field 
windings  by  the  rotating  magnetism  is  exactly  the  same 
as  though  the  windings  moved  with  an  equal  angular 
velocity  in  a  stationary  uniform  field.  The  electrical 
pressure  developed  in  a  conductor  depends,  in  other 
words,  upon  the  relative  angular  velocity  of  conductor 
and  magnetism,  and  it  makes  no  difference  which 
moves.  Therefore,  the  formula  at  the  bottom  of  page 
80,  Vol.  L,  applies  to  this  case,  or 


s 
IO8  X  60 


sm 


in  which  e1  is  the  instantaneous  pressure  in  the  coil,  ;/ 
the  number  of  turns  in  the  coil,  N  the  total  magnetiza- 
tion from  one  pole,  V  revolutions  per  minute,  and  a 
angular  position  of  the  coil.  In  the  case  under  consid- 
eration, 

AT=  22  ^rlBm  _  2  rlBm 

7T         2p  p 


*  Blondel,  Notes  sur  la  theorie  elementaire  des  appareils  a  champ 
tournant,  La  Lumiere  £lectrique,\To\.  50,  p.  351;  Jackson,  Three-phase 
Rotary  Field,  Electrical  Journal,  Vol.  I,  p.  185. 

2S 


626  ALTERNATING   CURRENTS. 

where  r  and  /  are  the  inner  radius  and  the  length  of  the 
field  core,  Bm  the  maximum  magnetic  density  in  the  air 
space,  and  p  the  number  of  pairs  of  poles  in  the  field. 

V      f 
Since  —  =s—  ,  and  the  magnetism  per  magnetic  circuit 

in  a  multipolar  machine  must  be  multiplied  by  the  num- 
ber of  pairs  of  poles  to  get  the  total  number  of  lines 
of  force  cut  per  conductor  per  revolution  (Vol.  L, 
p.  278),  the  formula  may  be  written,  generally,  in  the 
form 

,       2-irn'Nf   . 
e1  =  -  o^—  sin  a  ; 
I08 

2  irn'Nf 

whence  *_*«»  —    —  T~-> 

io8 


_,, 

and  E'  = 


R 
io8 


This  is  the  value  of  the  electrical  pressure  developed 
in  a  narrow  coil,  but  if  the  coil  is  spread  over  a  con- 
siderable area,  the  maximum  pressure  is  less  than  that 
given  above,  since  the  density  of  the  magnetism  which  is 
cut  by  the  conductors  falls  off  from  Bm  at  the  centre  of 
the  coil  to  some  smaller  value,  which  depends  upon  the 
width  of  the  coils.  If  the  coil  occupies  180  electrical 
degrees  of  circumference,  when  the  middle  conductor  is 
in  a  field  of  density  Bm  the  outer  conductors  are  in  a 
field  of  zero  density.  The  maximum  pressure  developed 
by  the  total  coil  is  then 

2      2-irn'Nf 

X  5  * 

7T  IO8 


ALTERNATING-CURRENT   MOTORS.  627 

If  the   coil   spreads  over  an  angle  0,  the  value  of  ej 

becomes 

.       I       2  wn'Nf  C^  , 

e  '  =  •£  x  -    — s-^-  I     cos  ada. 
6          io8     J-J0 

The  field  windings  of  induction  motors  are  usually 
arranged  uniformly  on  the  field,  so  that  in  two-phase 
motors  6  =  90°,  and  in  three-phase  motors  0  =  60°.  The 
respective  values  of  ej  become 


,  _  4V2  n'Nf 

IO8 

and 

i7  t  ^  3^2  n'Nf 
^  *      ~^~ 

Hence  the  electrical  pressure  set  up  in  a  uniformly 
distributed  field  winding  of  a  two-phase  motor  is,  other 
things  being  equal,  about  io  per  cent  less  than  if  the 
windings  were  in  narrow  coils;  and  in  a  three-phase 
motor  the  deficit  is  nearly  5  per  cent.  The  exact  ratios 
are  TT  :  2 A/2  and  TT  :  3.  To  give  the  same  counter  electric 
pressure  in  the  uniformly  distributed  field  windings  of 
an  induction  motor  arranged  for  two  phases,  requires 
about  6  per  cent  more  turns  in  the  windings  than  when 
the  same  machine  is  arranged  for  three  phases.  If-  the 
windings  are  placed  on  salient  poles,  as  is  sometimes 
done  in  two-phase  motors,  all  the  lines  of  force  pass 
through  the  windings,  and  the  coils  therefore  act  as 
though  they  were  very  narrow,  but.  the  increased  reluc- 
tance of  the  magnetic  circuit  caused  by  this  construction 
more  than  destroys  any  advantage  pertaining  to  the 
form  of  the  winding. 


628         ALTERNATING  CURRENTS. 

173.  Fo'rmulas  derived  from  Transformer  Formulas.  — 

In  the  case  of  a  transformer,  the  following  formula 
(Sect,  in)  gives  the  relation  between  electrical  press- 
ure, frequency,  magnetism,  and  the  turns  of  the  coils : 

EY  _  V2  irn'Nf 

£L ) 

I08 

provided  all  the  magnetism  is  included  within  all  the 
turns.  This  proviso  is  not  true  in  the  ordinary  induc- 
tion motor,  since  the  magnetic  density  in  the  air  gap 
may  be  assumed  to  vary  as  a  sinusoid,  and  that  condi- 
tion requires  that  the  number  of  lines  of  force  passing 
through  the  different  turns  of  the  coils  shall  also  vary 
as  a  sine  function.  This  is  illustrated  in  Fig.  290.  Con- 
sequently we  have  for  the  induction  motor, 


EV 

E'  = — — -J-  I     cos  ada, 


n 
where  -  is  the  value   of  a  corresponding  to   the  sine 

ordinate  which  is  proportional  to  the  number  of  lines 
of  force  passing  through  the  extreme  turns,  when  the 
total  magnetism,  Ny  passes  through  the  centre  turns  of 
the  coil. 

For  uniformly  distributed  windings  in  two  phases, 
6  —  90°,  and  in  three  phases,  6  =  60°,  while  for  coils  on 
salient  poles  0  =  o°.  The  values  of  E'  for  the  field 
winding  of  the  induction  motor  become 


E.,      4n'Nf        A     rr 
E<1  =  *  -  f£-    and    EJ  = 

io8  io8 

and  the  formulas  thus  developed  from  the  fundamental 
theorems  of  the  transformer  are  exactly  the  same  as 


ALTERNATING-CURRENT    MOTORS. 


629 


those  developed  from  the  conception  of  the  rotary  field. 
For  various  reasons  it  is  more  convenient  to  study 
induction  motors  from  the  transformer  standpoint,  and 
we  may  consider  them  as  transformers  with  a  relative 


Fig.  290 

motion  between  the  primary  and  secondary  windings. 
In  this  case,  the  field  winding  is  always  the  primary 
circuit  and  tJie  armature  winding  the  secondary  circuit. 

174.    Exciting  Current.  —  The  exciting  current  for  an 
induction  motor  may  be  calculated  for  each  circuit  in 


630         ALTERNATING  CURRENTS. 

exactly  the  same  manner  as  that  for  a  transformer.     It 
is  composed  of  two  components  in  quadrature  : 

(1)  The  active  current,  which  is  equal  to  the  sum  of 
the  no-load  losses  in  the  circuit,  in  watts,  divided  by  the 
volts  per  circuit.      The  number  of  circuits  is  equal  to 
the  number  of  phases.     The  total  losses  entering  into 
the  exciting  current  (or  the  no-load  losses)  are  the  core 
losses  in  the  field  and  armature;    the  C2R  loss  in  the 
field,  due  to  the  exciting  current  ;    a  small  C*R  loss  in 
the  armature,  due  to  the  armature  current  required  to 
run  the  armature  against  friction  and  core  losses  (usu- 
ally negligible)  ;  and  the  friction  loss.    The  watts  repre- 
sented in  the  exciting  current  per  electrical  circuit  are 
equal  to  the  total  no-load  losses  divided  by  the  number 
of  phases. 

(2)  The   wattless   magnetizing   current,  which  is  in 
quadrature  with  the  active  component  of  the  exciting 

current,  is  calculated,  as  in  the  case  of  transformers, 

_     P£[ 
from  the  formula  V5  nC  —  -  >  where  P  is  the  reluc- 

M      1.25 
tance  of  the  magnetic  circuit.     By  Section  160,  if  n'p  is 

number  of  turns  per  phase  which  link  each  magnetic 
circuit,  the  actual  magnetizing  current  per  circuit  is 

P*N  2 


which  for  a  two-phase  machine  becomes 

r  '-     PN 

C*-T^P' 

and  for  a  three-phase  machine, 

PN 


ALTERNATING-CURRENT   MOTORS.  631 

Since  mechanical  clearance  between  the  armature  and 
fields  is  an  essential  feature  of  a  motor,  the  reluctance 
of  the  motor  magnetic  circuit  is  much  greater  than 
that  of  a  transformer  of  corresponding  capacity.  This 
makes  the  magnetizing  current  greater,  increases  the 
exciting  current,  and  reduces  the  no-load  power  factor. 
The  total  exciting  current  is  equal  to  the  square  root 
of  the  sum  of  the  squares  of  its  two  components,  or 


175.  Field    Ampere-Turns.  —  The    ampere-turns    on 
each  magnetic  circuit  of  an  induction  motor  are  the 
resultant  of    the  ampere-turns  due  to    all   the   phases 
(Sect.  1  60).     It  may  readily  be  shown  that  the  result- 
ant of  any  number  of  equal  sine  functions  with  a  uni- 

form phase  difference  is  a  circular  function  equal  to  —  x, 

where  x  is  the  amplitude  of  the  components,  and  m 
the  number  of  components.  It  is  therefore  evident, 
from  the  deductions  of  Section  160,  that  the  resultant 
ampere-turns  in  the  magnetic  circuit  of  an  induction 

motor  is  ~(nfpC'V2),  where  n'p  is  the  number  of  turns 
belonging  to  each  phase  which  link  each  magnetic  cir- 
cuit, and  C'  is  the  current  in  each  phase.  Consequently, 
if  y  ampere-turns  are  required  in  the  magnetic  circuit, 

the  winding  in  each  phase  must  furnish  —  times  the 

2  m 

whole.       For  two-phase  motors,  —  =1,  and  for  three- 

,  22 

phase  motors,  —  =  -. 
m      3 

176.  Slip  and  Armature  Pressure.  —  As  stated  in  Sec- 
tion 165,  if  the  field  rotation  is  V,  the  armature  rotation 
must  be  somewhat  less,  or  V  ,  and  therefore  the  slip  is 


632         ALTERNATING  CURRENTS. 

V  —  V  =  v.  It  is  evident  that  v  is  proportional  to  the 
frequency  with  which  the  tgtal  useful  magnetism  per 
pole,  Na,  cuts  the  armature  conductors,  and  it  will  vary 
from  a  value  v  =  V  to  v  =  o  as  the  motor  armature 
changes  from  a  condition  of  rest  to  a  speed  of  synchro- 
nism. Slip  is  frequently  named  in  per  cent  of  motor 
speed.  In  ordinary  practice,  the  slip  varies,  at  full  load, 
from  2  per  cent  to  10  per  cent  of  V,  having  the  smaller 
value  in  large  machines.  The  maximum  pressure  in- 
duced in  any  conductor  on  the  armature  is 

ff  _2  7rNa  v^  _  2  TrNapv 
IO8        ~  IO8  X  60' 

where  Na  is  the  armature  magnetism  per  pole  ;  and  the 
effective  pressure  per  conductor  is 


io8  x  60 

since  the  pressure  curves  in  the  conductor  must  be  sinu- 
soidal if  the  magnetism  has  a  sinusoidal  distribution  in 
the  air  space,  as  has  been  assumed.  The  effective  cur- 
rent flowing  in  a  conductor  of  a  squirrel-cage  armature 
will  be 


_  a  a 

~"~     8" 


io8x6ox/c 

where  Ic"  is  the  impedance  of  the  armature  conductor, 
<j>a  is  the  angle  of  lag  of  the  current,  and  R"  the  resist- 
ance of  the  conductor.  The  C^R  loss  in  the  arma- 
ture conductors  will  be  Wa=S"CJ'zR^',  where  S"  is 
the  number  of  armature  conductors.  The  torque  is 
proportional  to  the  current  times  the  magnetic  field 
(which  is  nearly  constant);  and,  if  we  neglect  the 


ALTERNATING-CURRENT   MOTORS.  633 

effect  of  magnetic  leakage,  it  is  evident  that  the  arma- 
ture must  run  at  a  slip  which  sets  up  an  electric  pressure 
in  the  armature  conductors  which  is  equal  to  the  drop  of 
pressure  in  the  conductors  caused  by  the  current  de- 
manded to  give  the  torque.  So  that  the  slip  is  directly 
proportional  to  the  armature  current  for  a  fixed  arma- 
ture resistance,  and  for  a  fixed  armature  current  is 
directly  proportional  to  the  armature  resistance.  In 
actual  motors,  magnetic  leakage  is  not  negligible,  and 
the  slip  is  increased,  since  the  magnetic  leakage  in  a 
transformer  is  proportional  to  the  secondary  ampere- 
turns.  The  slip  is  also  increased  by  the  drop  of 
pressure  in  the  field  winding,  which  increases  with 
the  load,  and  causes  a  corresponding  decrease  in  the 
value  of  Na. 

The  fact  then  stands  that  the  increase  of  slip  between 
no-load  and  full-load  must  be  sufficient  to  increase  the 
pressure  in  the  armature  conductors  by  an  amount  equal 
to  the  increased  loss  of  pressure  in  the  conductors  due 
to  the  increased  current  and  the  decreased  armature 
magnetism.  If  magnetic  leakage  increases  with  the 
armature  current,  the  total  slip  becomes  proportional  to 


t  +       CJRJ,  where  A  is  a  constant,  Nt  the 
o 

number  of  leakage  lines  of  force  passing  through  the 
armature  coils,  S',  S"  the  number  of  field  and  armature 
conductors,  Ce'Re'  the  drop  of  pressure  per  field  con- 
ductor. 

177.  Design  of  Induction  Motors.  —  The  general  prin- 
ciples of  transformer  design  may  be  so  applied  to  the 
induction  motor  that  its  construction  becomes,  in  many 


634  ALTERNATING   CURRENTS. 

respects,  the  same  as  that  of  the  transformer.  If  it  is 
desired  to  design  a  rotary-field  motor  to  supply  W 
horse  power,  or  W  =  W  x  746  watts,  the  formula 


will  give  the  product  of  the  field  turns  into  the  mag- 
netism, n' N;  E'  and  /  being  given  by  the  conditions 
of  the  problem,  and  p  the  number  of  pairs  of  poles 
(which  is  dependent  upon  armature  speed  and  fre- 
quency) being  chosen  from  the  table  in  Section  170. 

2  A/2 

K  is  a  constant  which   is   equal   to   -     —  =  .90  for  a 

•?  7r 

two-phase    machine,    and    —  =  .95    for    a    three-phase 

7T 

machine  (Sect.  172).  It  must  be  remembered  that  E' 
is  the  primary  pressure  per  coil  in  each  phase,  and  its 
relation  to  the  line  pressure  per  phase  depends  upon 
the  connections  of  the  coils. 

The  safe  circumferential  speed  for  induction  motors 
is  even  higher  than  that  for  alternators,  since  the  con- 
ductors are  nearly  always  embedded  in  both  field  and 
armature  cores,  but  the  usual  periphery  velocities  are 
between  4000  and  6000  feet  per  minute.  With  a  given 
frequency  and  number  of  revolutions  per  minute,  the 
armature  diameter  in  feet  will  be 

U 


where  U  is  circumferential  speed  in  feet  per  minute. 
The  ratio  between  n1  and  N  must  be  determined  in  a 
great  measure  by  practice.  A  reasonably  large  value 


ALTERNATING-CURRENT   MOTORS. 


635 


of  N  with  a  proportionate  iron  section  increases  the 
no-load  losses,  but  increases  the  full-load  efficiency  as 
in  the  case  of  a  transformer  (Sect.  129).  It  is  usually 
desirable  to  have  the  maximum  efficiency  of  a  motor 
occur  at  about  three-fourths  load,  as  motors  commonly 
run  on  loads  which  average  less  than  full  load.  Kolben  * 
states  that  for  ordinary  good  practice,  the  number  of 
ampere-turns  per  centimeter  of  the  field  circumference 
at  full  load  should  be  from  100  to  150,  when  the  fre- 
quency is  from  40  to  80  and  the  induction  in  the  air 
gap  from  2000  to  3000.  This  should  only  be  used  as 
a  guide  or  check  in  making  a  design.  The  number  of 
field  turns  per  volt  appears  from  the  examination  of  a 
limited  number  of  machines  to  be  more  than  double 
the  number  of  turns  per  volt  used  in  transformers 
(p.  532);  the  range  being  from  —  ^  to 


100 


when  the  output  is  in  watts.  ^output      Voutput 

The  magnetic  density  in  the  field  cores  may  be  about 
the  same  as  in  transformers.  Kolben  gives  these  val- 
ues for  different  frequencies. 


/ 

B 

f 

B 

40 

5° 
60 

5500  to  6500 
5000  to  6000 
4500  to  5000 

80 
IOO 
120 

4000  to  4500 
4000  to  3500 
3500  to  3000 

Having  the  total  magnetization  N,  which  is  obtained 
from  the  pressure  formula  when  n'  is  assumed,  and 
assuming  the  maximum  magnetic  density  Bm,  the  cross- 

*  Electrical   World,  Vol.   22,   p.   284;    London    Electrician,  Vol.    31, 
P-  591. 


636  ALTERNATING   CURRENTS. 

section  of  the  iron  may  be  found.  The  magnetic  density 
between  the  core  slots  may  be  allowed  to  become  as 
great  as  two  or  two  and  a  half  times  the  density  in  the 
core,  but  every  effort  to  keep  it  small  in  value  should 
be  made. 

There  must  be  n'  turns  in  the  coils  of  each  phase, 
since  the  counter  pressure  E'  must  be  generated  in  the 
coils  of  each  phase.  The  length  of  field  or  armature 
will  be  dependent  upon  the  magnetic  density  in  the 
air  space,  which  may  be  made  quite  large.  The  limit 
being  determined  by  the  reluctance  permissible  in  the 
magnetic  circuit  and  hence  by  the  magnetizing  force 
required  to  drive  the  magnetism  through  the  circuit. 
The  air  space  of  the  motor  may  be  very  small,  so  that 
it  may  be  permissible  to  run  the  maximum  magnetic 
density  somewhat  higher  than  in  the  case  of  dynamos 
or  alternators,  though  the  practice  for  these  machines 
may  be  safely  followed.  From  2000  to  6000  lines 
would  be  a  safe  range.  If  the  fields  are  wound 
through  holes,  as  in  Fig.  287,  the  length  of  the  field 
will  be 

TT      2pN    _pN 
~ 


where  p  is  the  number  of  pairs  of  poles,  Bm  the  maxi- 
mum air-space  induction,  TrD  the  circumference  of  the 
polar  surface,  and  N  is  the  magnetism  emanating  from 
a  pole.  The  paths  of  leakage  are  somewhat  as  shown 
in  Fig.  291,  and  the  coefficient  of  leakage  may  be 
taken  between  1.05  and  1.25  when  the  motor  is  run- 
ning at  full  load.  At  starting,  the  leakage  will  be 
increased  on  account  of  the  strong  armature  reaction 


ALTERNATING-CURRENT   MOTORS.  637 

tending  to  force  the  field  magnetism  across  the  pole 
tips.  The  magnetizing  current  for  a  two-phase  machine 
may  be  found  from  the  formula  (Sect.  174) 

C  '  =     PN    =   $6  —  , 
**        1.7^  nJ  nJ 


where  P   is   the   reluctance   of  the 
magnetic  circuit.     For  three-phases 

C.  '  =    PN    =     8 PN 

2.65  nff  nf'  Fig.  291 

The  working  current  in  each  phase  of   the  two-phase 
winding  is 

C2J  =  ~%=j-  -T-  per  cent  efficiency. 

The  efficiency  of  the  motor  is  assumed  during  the  trial 
calculation.     In  the  three-phase  winding 

C,J  =  ^=-r  -*-  per  cent  efficiency. 

JL> 

The  total  current  in  a  two-phase  machine  is 


J=^C,J*+C^\ 


and  for  the  three-phase, 


Having  obtained  the  field  current,  the  size  of  the  field 
conductors  may  be  obtained,  allowing  from  900  to  1200 
circular  mils  per  ampere. 

As  in  transformers,  the  radiating  surface  of  a  station- 
ary field  core  should  not  be  less  than  five  to  seven  square 


638 


ALTERNATING    CURRENTS. 


inches  per  watt  radiated.  If  the  field  rotates,  this  may 
be  greatly  reduced.  The  usual  way  to  wind  the  coils  is 
to  divide  them  up  into  sections  and  place  them  in  slots 
or  holes  (Fig.  292).  Slots  seem  to  be  preferable  if  the 
teeth  are  close  together,  as  there  is  then  less  field  leakage 
in  the  paths  indicated  by  the  dotted  lines  of  Fig.  292  #. 
The  armature  may  be  wound  in  any  of  the  ways 
explained  in  Section  169.  The  wires  are  usually  em- 


Fig.  292 

bedded  in  the  surface  of  the  core,  as  is  the  case  with  the 
field  windings  (see  Figs.  287  and  288).  By  this  means 
the  air  space  may  be  made  of  exceedingly  small  depth, 
and  the  magnetizing  current  and  magnetic  leakage  are 
thus  reduced,  which  is  very  desirable.  The  diameter  of 
the  armature  has  already  been  determined ;  its  speed  will 
be  V  =  V  —  v.  The  slip,  v,  should  be  made  from  2  to 
10  per  cent,  the  larger  value  being  for  a  machine  of 


ALTERNATING-CURRENT   MOTORS.  639 

about  i  H.P.  and  the  lower  for  100  H.P;*  in  other 
words,  the  regulation  of  induction  motors  may  be  made 
equal  to  that  of  continuous-current  motors.  The  copper 
loss  in  the  armature  bars  Lr"  is 


Lc"  = 
c,,Rn  = 


IO    X  OO 


,  ,,  . 

io8  x  60  x  Ce" 

In  a  squirrel-cage  armature  K  =  i,  but  in  coil  armatures 

i  C*e 
its  value  depends  upon  -  I     cos  ada  (Sect.  172).     The 


value  of    Cc"  is  given  with  ample  accuracy  from   the 

formula    W=  ~EfS"Cc"  cos  <£",  where  Sf  and  S"  are 
o 

respectively   the   number  of   embedded    conductors   of 

field  and  armature.  If  both  field  and  armature  are 

eft  nii 

either  drum  or  ring  wound,  it  is  evident  that  -—  =  —-, 

S'  n 

but  if  they  have  different  types  of  windings  the  equality 

Su      E" 
does  not  exist.     In  every  case  —  =  — •    The  value  of 

vj  J~^f 

cos  0"  may  be  taken  as  0.90  for  a  reasonably  close 
approximation,  and  the  value  of  the  resistance  of  each 
armature  conductor,  including  its  share  of  the  resistance 
of  the  end  connections,  which  corresponds  to  a  fixed 
value  of  the  slip,  is  then  approximately  determined 
from  the  formula.  The  value  of  Na  used  in  this  com- 

E1  —  C'  R' 
putation  must  correspond  to  full  load ;  it  is  N— '- — 


*  Kapp's  Electrical  Transmission  of  Energy,  4th  ed.,  p.  323;  Steinmetz, 
Trans.  Am.  Inst.  E.  E.,  Vol.  n,  p.  37. 


640  ALTERNATING   CURRENTS. 

where  N  is  the  field  magnetization  at  no  load  (that  is, 
assuming  no  CR  drop  in  the  primary),  E-  is  the  im- 
pressed circuit  pressure,  O  is  the  primary  current  at  full 
load,  and  z  is  the  leakage  coefficient. 

If  the  armature  is  not  arranged  to  allow  the  insertion 
of  a  starting  resistance,  and  if  a  maximum  starting 
torque  is  desired,  R"  must  be  made  of  such  a  value  as 
to  make  Rc"  =  2  w/Z,/',  where  Lc"  is  the  self-inductance 
of  an  armature  bar.  Such  a  value  for  R"  gives  an 
enormously  large  slip  at  full  load  and  is  unsatisfactory 
except  for  special  purposes,  so  that  Rc"  is  usually  smaller. 

The  density  of  current  in  the  armature  conductors,  if 
the  armature  rotates,  may  be  made  quite  large,  an 
allowance  of  300  circular  mils  per  ampere  being  suffi- 
cient, since  the  core  losses  are  insignificant.  If  the 
armature  is  stationary,  the  current  density  should  not 
exceed  one-third  that  value.  The  radiating  surface  per 
watt  lost  in  a  rotating  armature  should  be  the  same  as 
that  in  an  alternator  or  continuous-current  dynamo, 
taking  all  losses  into  account.  In  the  same  way  the 
radiating  surface  of  a  stationary  armature  should  be  the 
same  as  is  allowed  for  dynamo  fields  if  the  same  rise  of 
temperatures  is  admitted. 

The  foucault  current  and  hysteresis  losses  may  be 
determined  exactly  as  in  the  case  of  a  transformer, 
using  v^  as  the  frequency  of  magnetic  cycles  in  the 
armature,  and  /  as  the  frequency  in  the  field. 

The  loss  in  primary  pressure  due  to  C^R  losses  in  the 
fields,  and  foucault  currents  and  hysteresis,  will  increase 
proportionally  the  input  required  to  give  a  desired  out- 
put, and  proper  correction  must  be  made  in  the  design. 


ALTERNATING-CURRENT   MOTORS.  641 

The  efficiency  of  a  machine  is 

W 


W+L 


77  = 


and  L=H,+  Ha  +  Z,  +  Za  +  C*Rf  +  C"2Ra  +  F,  where 
W  is  the  output,  L  the  total  losses,  H  hysteresis  losses, 
Z  f  oucault  current  losses,  and  F  friction  losses,  all  given 
in  watts  or  horse  power.  The  maximum  efficiency  evi- 
dently occurs,  as  in  continuous-current  machines  and 
transformers,  at  that  load  which  causes  the  variable 
copper  losses  to  equal  the  constant  core  and  friction 
losses. 

The  power  factor  of  the  machine  when  running  with- 
out load  is 


as  shown  in  Section  174.  When  the  machine  is  loaded, 
the  power  factor  is  partially  dependent  upon  the  lag  of 
current  in  the  armature  (cos  $"),  and  the  self-induc- 
tances of  both  armature  and  field  windings  must  be 
calculated  before  the  power  factor  can  be  determined. 
The  self-inductances  of  the  windings  are  due  to  the 
leakage  lines  of  force,  and  the  values  may  be  determined 
from  the  reluctances  of  the  leakage  paths  and  the 
arrangements  of  the  windings.*  In  general,  the  power 
factor  is  approximately  equal  to 


*  London  Electrician,  Vol.  36,  p.  578. 

2T 


642         ALTERNATING  CURRENTS. 

The  torque  of  the  armature  when  the  output,  W,  is 
given  in  watts  is  equal  to 


— — x  IO    in  dyne-centimeters, 

27rF' 

Wx  io7 
: in  gramme-centimeters, 

2irV  i6.3 


.3 
W  x  io7 


2  TT  V  226,000 


in  pound-feet. 


177  a.  Output  Proportional  to  Square  of  Primary  Press- 
ure. —  The  output  of  the  motor,  plus  the  armature-core 
losses  and  friction,  is  equal  to  the  product  of  the  number 
of  armature  conductors  and  the  effective  current  in  each 
conductor,  multiplied  by  the  product  of  the  pressure 
which  would  be  developed  in  each  conductor  if  the 
armature  were  driven  at  its  speed  in  an  equal  stationary 
field  and  the  cosine  of  the  angle  of  lag  of  the  armature 

current, 

nr,,V     ,,          ,,, 

v 


but  C."  = 

and  cos  <£"  = 

and  therefore         W  = 
Also  E."*  = 

1/6  ^  I  &  I/'  A 

where  El  is  the  induced  field  pressure  per  conductor,  E' 


— 

v 


ALTERNATING-CURRENT   MOTORS.  643 

the  total  field  pressure,  and  Sf  the  number  of  field  con- 
ductors, 

S"E,*V*  Ru  (S" 

~s*     ~*    c 

W= 


This  formula  is  not  one  which  can  be  made  of  service 
in  the  design  of  a  motor  (in  fact,  it  is  not  needed  for 
such  a  purpose),  but  it  plainly  shows  the  effect  on  the 
output  of  a  motor,  which  is  caused  by  varying  any 
one  of  its  constructive  details  while  the  others  remain 
unchanged.  A  very  important  deduction  from  the  for- 
mula is  :  that  the  torque  and  output  of  an  induction  motor 
vary  as  the  sq2iare  of  the  primary  pressure,  so  that  a 
machine  which  will  carry  an  overload  of  50  per  cent  on 
its  normal  pressure  will  barely  run  at  full  load  if  the 
pressure  is  reduced  20  per  cent.  The  formula  also 
shows  that  the  slip  is  inversely  dependent  on  the  pri- 
mary pressure. 

178.  Electromagnetic  Repulsion.  —  If  a  coil  of  wire  is 
held  in  an  alternating  magnetic  field  in  such  a  way  that 
the  lines  of  force  pass  through  its  turns,  an  alternating 
pressure  is  set  up  in  it  which  has  90°  difference  of  phase 
from  the  alternating  magnetism.  This  in  turn  causes  a 
current  in  the  coil,  and  the  coil  experiences  a  force  at 
each  instant  tending  to  move  it  in  the  magnetic  field, 
which  is  proportional  in  magnitude  and  direction  to  the 
product  of  the  corresponding  instantaneous  values  of 
current  and  magnetism,  paying  due  attention  to  their 
relative  signs  ;  and  the  force  for  a  period  is  equal  to 
the  average  of  the  instantaneous  torques  during  the 
period.  If  the  coil  could  have  no  self-inductance,  and 


644  ALTERNATING   CURRENTS. 

the  phase  of  the  current  could  therefore  be  in  quadra- 
ture with  that  of  the  magnetism,  the  average  forces 
during  alternate  quarter  periods  would  be  equal,  but  in 
opposite  directions  (compare  Fig.  49),  and  the  average 
force  during  a  whole  period  would  be  zero,  so  that  the 
coil  would  have  no  tendency  to  move ;  but  in  all  prac- 
tical cases  a  coil,  or  even  a  flat  disc,  must  have  some 
self-induction,  so  that  the  current  lags  behind  the  im- 
pressed pressure,  and  the  current  phase  is  therefore 
more  than  90°  behind  the  phase  of  the  magnetism. 
In  this  case  the  instantaneous  values  of  the  force,  when 
plotted  in  a  curve,  give  a  figure  similar  to  Fig.  48,  but 
turned  upside  down,  since  the  current  lags  behind  the 
magnetism  more  than  90°.  The  ordinates  of  the  large 
loop  represent  a  negative  or  repulsive  force,  and  the 
ordinates  of  the  small  loops  a  positive  or  attractive 
force,  and  the  summation  of  the  instantaneous  forces 
during  a  period  is  seen  to  have  a  finite  negative  value. 
This  shows  that  the  coil  experiences  a  repulsive  force 
which  tends  to  move  it  out  of  the  magnetic  field.  If 
the  coil  is  pivoted,  the  force  tends  to  turn  it  into  such 
a  position  that  the  lines  of  force  of  the  field  do  not 
thread  through  its  turns.  The  conditions  here  set  forth 
were  first  fully  explained  and  illustrated  in  a  remarka- 
ble lecture  by  Professor  Elihu  Thomson,*  and  a  similar 
lecture  by  Professor  Fleming,  f 

179.   Single-Phase  Induction  Motors.  —  If  the  fields  of 
an  induction  motor  are  wound  with  one  set  of  coils  so 

*  Novel  Phenomena  of  Alternating  Currents,  Trans.  Anier.  Inst.  E.  E., 
Vol.  4,  p.  1 60. 

f  Electromagnetic  Repulsion,  London  Electrician,  Vol.  26,  p.  567. 


ALTERNATING-CURRENT   MOTORS. 


645 


that  the  field  poles  are  set  up  by  a  single  alternating 
current  flowing  in  the  coils,  the  poles  will  be  stationary 
but  alternating,  and  the  effects  of  electromagnetic  repul- 
sion just  described  may  be  utilized  for  the  purpose  of 
causing  the  armature  to  rotate.  If  the  armature  is 
wound  with  uniformly  spaced  short-circuited  coils  or 
conductors,  the  repulsive  effects  in  the  different  coils 
will  balance  each  other  when  the  armature  stands  still ; 


Fig.  203 

but  if  the  coils  have  their  independent  ends  separately 
connected  to  the  opposite  bars  of  a  commutator  having  as 
many  bars  as  there  are  sets  of  conductors  in  the  arma- 
ture, brushes  may  be  so  arranged  as  to  short-circuit  each 
coil  when  it  is  in  a  position  to  give  a  force  in  one  direc- 
tion. This  arrangement  was  suggested  by  Professor 
Thomson  *  and  is  illustrated  in  Fig.  293.  The  motor 
is  self-starting,  and  runs  by  virtue  of  the  repulsion 

*  Trans.  Amer.  Inst.  E.  £.,  Vol.  4,  p.  160. 


646  ALTERNATING   CURRENTS. 

between  the  magnetic  field  and  the  coils  which,  as 
they  come  into  the  active  position,  are  short-circuited 
by  the  brush  connections.  Such  a  motor  is  bulky,  in- 
efficient, and  expensive,  since  only  a  portion  of  the 
armature  can  be  made  continuously  effective ;  but  if  a 
uniformly  wound  short-circuited  armature  (such  as  is 
used  for  polyphase  induction  motors)  is  started  to  re- 
volving in  a  single-phase  alternating  magnetic  field,  the 
balance  of  repulsions  which  exists  when  the  armature  is 
at  rest  is  disturbed,  and  the  armature  tends  to  continue 
its  motion.  To  illustrate  this,  the  condition  of  two  coils 
in  complementary  positions  with  reference  to  one  of  the 
poles  may  be  considered.  As  the  armature  revolves, 
one  coil  moves  toward  a  position  where  it  includes  more 
lines  of  force  from  the  pole,  and  the  other  coil  moves  so 
as  to  exclude  lines  of  force.  If  the  strength  of  pole  is 
rising,  the  first  coil  will  have  the  larger  current  induced 
in  it,  since  the  rate  of  change  of  lines  of  force  through 
the  first  coil  is  equal  to  the  sum  of  the  rate  of  change 
in  the  strength  of  field  and  the  rate  at  which  the  coil 
moves  through  the  field,  while  the  rate  of  change  of 
lines  of  force  through  the  second  coil  is  the  difference 
of  these  two  rates.  Thanks  to  the  lag  in  the  coil  cir- 
cuits, the  currents  in  both  coils  are  in  such  a  direction 
as  to  result  in  an  attractive  force  on  the  poles,  but  a 
much  stronger  force  is  experienced  by  the  first  coil 
than  the  second.  When  the  field  is  falling,  the  mag- 
netic condition  of  the  second  coil  is  changing  most 
rapidly,  but  the  direction  of  the  induced  currents  in 
the  coils  is  reversed  with  respect  to  the  direction  of  the 
fields,  and  the  coils  experience  a  repulsive  force  with 


ALTERNATING-CURRENT   MOTORS.  647 

reference  to  the  pole.  The  effect  during  one  complete 
period  of  the  magnetism  therefore  tends  to  cause  the 
armature  to  rotate  in  the  same  direction  in  which  it  was 
started.  The  torque  is  a  maximum  when  the  positive 
product  of  current  and  magnetism  is  a  maximum,  which 
is  when  the  current  lags  behind  the  induced  pressure 
by  an  angle  between  45°  and  90°.  The  torque  at  any 
speed  (slip,  v  =  V—  V)  is  equal  to  the  torque  which 
would  be  given  by  a  polyphase  induction  motor  of 
similar  construction  at  that  speed,  minus  the  torque 
which  the  polyphase  induction  motor  would  give  in 
a  field  of  double  the  frequency,  and  with  a  slip  equal  to 
F+  V  =  ?•  V  —  v\  but  with  the  ordinary  ratio  of  the 
resistance  and  inductance  in  the  armature  winding,  the 
torque  due  to  the  latter  slip  is  negligible  at  such  full-load 
slips  as  are  satisfactory  in  practice.  In  this  case,  V 
need  not  be  looked  upon  as  a  speed  of  rotation  of  a 
magnetic  field,  but  as  the  speed  of  the  armature  which 
keeps  each  conductor  at  the  same  position  with  refer- 
ence to  a  pole  for  any  fixed  instant  in  the  period  of  the 

magnetism.     Hence  V=  — -,  exactly  as  in  rotary-field 

machines.  When  the  armature  is  stationary,  v  =  V, 
and  the  two  torques  are  equal  and  opposite.  A  single- 
phase  induction  motor  may  therefore  be  designed  in 
exactly  the  same  manner  as  a  polyphaser  in  respect  to 
its  operation  after  it  has  reached  its  normal  speed,  but 
it  requires  special  treatment  in  the  design  for  the 
purpose  of  making  it  self-starting. 

180.   Resolution  of  Alternating  Field. —  The    single- 
phase  alternating  field  may  be  treated  in  a  different 


648 


ALTERNATING   CURRENTS. 


manner  to  get  exactly  the  same  result.  An  alternating 
field  stationary  in  position  may  be  resolved  into  two 
rotary  fields,  revolving  in  opposite  directions,  having 
the  same  frequency  as  the  stationary  field,  and  of  one- 
half  its  magnitude  or  strength.*  This  is  exactly 
similar  to  the  principle  of  mechanics  by  which  a  simple 
harmonic  motion  may  be  resolved  into  two  uniform 
opposite  circular  motions  of  one-half  the  amplitude. 


Fig.  294 

The  torque  diagrams  of  each  of  these  fields  acting  alone 
are  shown  in  Fig.  294,  where  O  is  the  point  of  armature 
rest,  and  armature  speed  is  counted  from  that  point 
along  the  horizontal  axis.  The  curves  A  and  Af  are 
the  torque  curves  that  would  be  given  by  either  field 
acting  alone,  the  torque  due  to  one  being  in  one  direc- 

*  Ferraris,  A   Method  for  the  Treatment  of  Rotating  or  Alternating 
Vectors,  London  Electrician,  Vol.  33,  p.  I IO. 


ALTERNATING-CURRENT   MOTORS.  649 

tion,  and  that  of  the  other  in  the  opposite  direction.  It 
is  evident  that  when  the  armature  is  at  rest  it  has  no 
tendency  to  revolve,  as  the  slip  of  the  armature  with  re- 
spect to  the  two  fields  is  equal,  and  torques  Ot  and  Ot' 
created  by  the  two  fields  are  equal  and  opposite;  but  if 
the  armature  is  started  in  one  direction,  for  instance 
toward  the  right,  the  slip  with  respect  to  field  A  decreases, 
the  torque  caused  by  it  increases  and  tends  to  continue 
the  rotation,  while  the  slip  with  respect  to  field  A'  in- 
creases, and  the  torque  caused  by  A'  decreases.  When 
the  armature  speed  becomes  V ,  the  torque  caused  by 
A  is  Tt  which  is  due  to  a  slip  V—  V  =  v ;  while  the 
torque  caused  by  Af  is  Tr,  which  is  due  to  a  slip  in 
relative  speed  between  armature  and  field  of 

F+  V  =  2  V-  v. 

From  the  relations  of  torque  to  slip,  which  have  already 
been  discussed  (Sects.  167  and  168),  it  is  evident  that 
the  torque  caused  by  A'  decreases  as  the  relative  speed 
increases  above  V,  If  the  differences  between  the 
corresponding  ordinates  of  the  curves  of  torque  due  to 
A  and  Af  are  plotted  in  a  curve,  the  actual  torque 
curve,  M,  is  given.  The  ordinates  of  this  will  give  the 
actual  motor  torque  with  respect  to  slip.  From  this 
curve  it  is  seen  that  the  motor  will  work  at  no  load  at 
an  almost  synchronous  speed,  and  may  then  be  loaded 
until  the  speed  has  dropped  to  a  point  where  the  torque 
is  at  a  maximum.  If  the  load  exceeds  this,  the  motor 
will  stop. 

If  the  armature  should  be   started  toward  the  left, 
instead  of  the  right  as  here   assumed,  the  conditions 


650  ALTERNATING   CURRENTS. 

would  be  reversed  and  the  motor  would  operate  under 
the  torque  line  Mf.  The  ratio  --  should  be  large  in  a 

single-phase  motor,  in  order  that  the  curve  A'  may 
be  close  to  the  X  axis  at  ordinary  running  speeds,  but 
the  ordinary  values  of  resistance  and  inductance  which 
are  required  in  an  efficient  and  economical  design  make 
the  effect  of  A'  so  small  at  the  speed  of  normal  full  load 
that  the  action  of  A  only  need  be  considered. 

181.  Formulas  for  Single-Phase  Induction  Motors.  - 
From  these  considerations  it  is  seen  that  a  single- 
phase  motor  may  be  designed  in  exactly  the  same 
manner  as  a  polyphaser,  and  that  for  equal  output  the 
resultant  ampere-turns  upon  the  field  must  be  equal  to 
the  number  on  a  polyphase  motor.*  The  field  wind- 
ing may  be  arranged,  as  in  polyphase  motors,  cover- 
ing the  entire  polar  surface,  as  is  shown  in  Fig.  295, 
for  a  four-pole  machine;  but  the  differential  action 
in  this  case  reduces  the  effectiveness  of  the  winding  in 
the  proportion  of  I  :  —  ,  as  has  already  been  shown  in 

7T 

Section  172.  Consequently,  the  equation  from  which 
the  field  windings  are  determined  becomes 


_ 


IO8  '      7T  IO8  IO8 

The  value  of  K  may  be  increased,  and  material  saved, 
by  leaving  space  between  the  coils  as  in  Fig.  296, 
which  shows  the  windings  for  a  two-pole  field. 

Not  only  may  the  armatures  of    single-phase   induc- 

*  Kolben,  Design   of    Alternating-Current   Motors,  Electrical    World, 
Vol.  22,  p.  284;   London  Electrician^  Vol.  31,  p.  590. 


ALTERNATING-CURRENT   MOTORS. 


65I 


tiori  motors  be  designed  in  exactly  the  same  way  as  are 
those  of  polyphasers,  but  an  armature  which  gives  the 


Fig.  295 

best  results  when  running  in  a  polyphase  field  is  likely 
to  be  best  for  a  single-phase  machine,  other  things  being 
equal. 


Fig.  296 


The  efficiency  of  single-phasers  should  be  slightly 
less  than  that  of  polyphasers,  since  the  armature-core 
losses  are  proportional  to  the  frequency  of  the  main 
field  instead  of  to  the  slip ;  their  slip  for  a  given  load 


652  ALTERNATING   CURRENTS. 

and  similar  design  is  slightly  greater,  and  their  maximum 
torque  is  slightly  less  than  that  of  polyphasers,  as  is 
shown  by  Fig.  294 ;  but  these  differences  in  well-designed 
machines  should  not  be  great.  The  weight  of  single- 
phasers  is  larger  than  that  of  equal  polyphasers,  because 
the  value  of  K  is  smaller. 

182.  Starting  Single-Phase  Induction  Motors.  —  Since 
single-phase  induction  motors  are  not  per  se  self-starting, 
special  starting  devices  must  be  included  in  the  design 
and  construction.     As  a  rule,  this  takes  the  form  of  what 
is  called  a  Phase  Splitter.     The  field  is  wound  with  two 
coils  similar  to  the  windings   of  a  two-phaser,  and  at 
starting  these  are  connected  in  parallel  to  the  circuit; 
one  directly,  and  the  other  through  a  dead  resistance  or 
capacity.     This  throws  the  current  in  the  two  coils  into 
a  difference  of  phase,  which  may  be   accentuated   by 
winding  one  coil  so  that  it  has  greater  self-inductance 
than  the  other ;  and  the  machine  then  starts  as  a  two- 
phaser.     After  the  machine  is  running,  the   coils    are 
connected  directly  to  the  circuit,  or  one  coil  is  cut  out, 
and  the  motor  operates  as  a  single-phaser.     The  opera- 
tion of  "phase  splitting,"  as  applicable  to  such  motors, 
cannot  give  a  large  difference  of  phase  between  the  cur- 
rents in  the  two  motor  circuits  with  a  reasonably  large 
power  factor,  and  consequently  single-phase  induction 
motors  must  have  either  a  very  small  starting  torque 
or  an  unreasonably  small  power  factor  at  starting. 

183.  Efficiency  of  Induction  Motors  and   Methods   of 
Making   Tests.  —  Polyphase   induction    motors  can    be 
built   with    about    the    same   efficiency    as    continuous- 
current  motors,  and  with  somewhat  less  cost  on  account 


ALTERNATING-CURRENT   MOTORS.  653 

of  the  absence  of  a  commutator  and  the  low  insulation 
required  on  the  armature  conductors,  but  with  a  counter- 
balancing extra  cost  on  account  of  the  high  grade  and 
expensive  sheet  iron  stampings  which  are  required  for 
the  field. 

1.  Direct  Measurement,     In  testing  the  efficiency  of 
these  motors,  the  output  may  be  measured  by  a  brake 
or  transmission  dynamometer,  as  is  explained  for  testing 
continuous-current  motors  in   Vol.   I.,  p.   255,   but   the 
input   must   be    measured    by    one    of    the    wattmeter 
methods  of  Section  44.     The  two-wattmeter  method  is 
the  best,  but  care  must  be  taken  to  determine  whether 
the  readings  are  additive  or  subtractive,  since  the  power 
factor  of  a  partially  loaded  induction  motor  is  likely  to 
be  quite  low  and  at  no  load  may  be  only  a  few  per  cent. 
The  power  factor  is  determined  by  taking  simultaneous 
readings  of  amperemeter,  voltmeter,  and  wattmeter  in 
one  circuit,  if  the  machine  is  balanced;  but  if  the  cir- 
cuits  differ,  readings  for  each  circuit  must  be   taken. 
The  power  factor  is  then  the  true  watts  divided  by  the 
apparent  watts.     This  method  requires  that  the  motor 
shall  be  operated  with  its  full  load,  and  therefore  may 
prove  inconvenient,  and  it  does  not  give  any  way  of 
separating  the  losses. 

2.  Stray  Power  Method.     A  method  similar  to  that 
described    for  testing   transformers   (Sect.    125,   8  a)  is 
often  more  convenient  and  satisfactory.     By  this  plan 
the  core  losses  and  friction  losses   are   determined  by 
measuring  by  wattmeter  the  power  which  is  absorbed 
by  the  motor  when  running  light  under  normal  pressure 
and  frequency.     As  the  power  factor  under  such  condi- 


654         ALTERNATING  CURRENTS. 

tions  is  likely  to  be  very  small,  the  field  current  flowing 
is  considerable  and  the  C2R  loss  cannot  be  neglected, 
but  a  correction  can  be  made  after  the  test  for  copper 
losses  is  completed.  To  measure  the  copper  losses,  the 
machine  is  locked  so  as  to  remain  stationary,  in  which 
case  the  armature  serves  the  purpose  of  a  short-circuited 
secondary,  and  such  a  reduced  impressed  pressure  is 
applied  as  to  cause  any  desired  current  to  flow  in  the 
fields.  The  wattmeter  readings  give  the  C2R  losses  for 
the  current  flowing,  and  the  losses  for  any  other  current 
may  be  at  once  calculated.  A  small  core  loss  is  included 
in  this  measurement,  but  should  commonly  be  negligible  ; 
an  approximate  correction  may  be  made  on  account  of 
it,  when  necessary,  by  considering  its  ratio  to  the  total 
corrected  core  losses  as  the  1.6  power  of  the  pressure 
applied  in  the  copper  loss  test  is  to  the  1.6  power  of  the 
normal  pressure.  From  these  results  the  total  losses 
and  the  efficiency  at  any  load  may  be  calculated. 

A  motor  running,  without  load,  or  with  part  load,  on 
an  unbalanced  circuit,  is  likely  to  absorb  widely  different 
amounts  of  power  in  its  coils ;  one  coil  may  even  return 
power  to  the  circuit,  while  the  others  absorb  the  power 
required  for  operation  plus  that  returned.  In  all  such 
cases,  the  two-wattmeter  method  of  measuring  the 
power  gives  the  net  power  absorbed  by  the  machine. 

3.  Power  Factor.  The  power  factor  at  any  load  may 
also  be  calculated  from  the  results  of  the  two  loss  tests. 
Thus,  from  the  power  factor  of  the  machine  running 
light,  which  is  determined  in  the  core-loss  tests,  is  read- 
ily deduced  the  wattless  magnetizing  current,  since  the 
power  factor  is  equal  to  cos  <//  and  the  magnetizing  cur- 


ALTERNATING-CURRENT   MOTORS.  655 

rent  per  coil,  CJ,  is  equal  to  C±  sin  (//,  where  C^  is  the 
current  per  coil  as  shown  by  the  amperemeter.  The 
field  current  at  any  load  is  equal  to  the  resultant  of 
the  active  current  corresponding  to  that  load  and  the 
wattless  current,  or 


where  C  is  the  field  current  per  coil,  E'  is  the  normal 
pressure  per  coil,  W  is  the  total  output,  L  is  the  total 
losses,  m  is  the  number  of  phases,  and  £T/  the  wattless 
magnetizing  current  per  coil.  The  power  factor  at  any 

' 


4.  Regulation  and  Torque.  —  All  desired  information 
in  regard  to  the  operation  of  induction  motors  (except 
regulation  and  starting  torque)  may  therefore  be  deter- 
mined by  purely  electrical  measurements,  and  to  a  high 
degree  of  accuracy.  Using  commercial  amperemeters, 
voltmeters,  and  wattmeters  which  have  been  properly 
calibrated,  the  errors  probably  affecting  the  full-load  effi- 
ciency and  power  factor  need  not  exceed  one  per  cent. 
The  exact  regulation  of  a  machine  can  be  determined 
only  by  actual  running  tests  under  load,  in  which  the 
t  actual  slip  is  measured,  but  the  percentage  slip  may  be 
taken  to  be  approximately  equal  to  the  percentage  drop 
of  pressure  in  the  windings.  The  starting  torque  can 
be  measured  by  clamping  a  lever  upon  the  pulley,  and 
measuring  the  pull  at  the  end  of  a  fixed  length  of  arm. 
For  machines  having  an  improperly  divided  starting  re- 
sistance in  the  armature,  this  gives  a  value  which  is 


656 


ALTERNATING   CURRENTS. 


higher  than  the  torque  against  which  the  motor  will  start 
and  run  up  to  speed.  In  such  machines,  the  standing 
torque  and  starting  torque  are  different,  but  in  machines 
having  a  properly  arranged  starting  resistance  in  the 
armature,  the  standing  torque  and  starting  torque  are 
equal,  and  are  practically  equal  to  the  maximum  torque 
which  the  machine  can  exert. 

The  following  table  gives  the  efficiency,  power  factors, 
and  other  characteristics  of  a  number  of  European  poly- 
phase motors.* 


Designation 

o 

I 

2 

Capacity  —  H.P   .     ... 

1 

2 

6 

24 

60 

Field  speed      

1500 

1500 

IOOO 

IOOO 

75° 

Speed  at  full  load      .     .     . 

1435 

1445 

960 

970 

730 

Number  of  phases     .     .     . 

3 

3 

3 

3 

3 

Frequency  .... 

50 

50 

50 

50 

50 

Pressure 

190 

190 

190 

190 

190 

No-load  current    .... 

2.2 

3-6 

12 

25 

48 

Full-load  current  .... 

3-3 

7-5 

20.5 

70 

1  60 

Full-load  efficiency    .     .     . 

68 

75 

82 

9i 

93 

Power  factor  at  full  load     . 

•75 

.80 

.80 

.85 

.90 

Number  of  poles  .... 

4 

4 

6 

6 

8 

Slip  at  full  load    .... 

4-3 

3-7 

4 

3 

2.7 

Designation     

5 

6 

7 

8 

9 

Capacity  —  H.P  .... 

IOO 

I25 

i 

5 

50 

Field  speed     

600 

500 

1500 

1500 

750 

Speed  at  full  load      .     .     . 

588 

488 

*375 

1395 

725 

Number  of  phases     .     .     .-/ 

3 

3 

3 

3 

3 

Frequency  

50 

50 

50 

50 

50 

Pressure 

IQO 

TOO 

IOO 

IOO 

IOO 

No-load  current   .... 

iy»j 

75 

iyu 

90 

4-5 

15 

150 

Full-load  current  .... 

265 

330 

8 

36 

280 

Full-load  efficiency    . 

93 

94 

75 

84 

9i 

Power  factor  at  full  load     . 

.91 

.91 

.70 

.70 

.82 

Number  of  poles  .... 

10 

12 

4 

4 

8 

Slip  at  full  load    .... 

2 

2.4 

8 

7 

3-3 

*  Thompson's  Polyphase  Electric  Currents,  p.  203;   Rodet  et  Busquet's 

Courants  Polyphases,  p.  89. 

ALTERNATING-CURRENT   MOTORS. 


657 


HORSE  POWER 


EFFIC  ENCY 


20          30  40         50  60          7.0          80          90         100        110 

Fig.  297 

TORQUE  IN  KILOGRAMS  ON  RADIUS  OF  EIGHT  CM. 
40 


160 


lOO 


10         20          30         40          50         60          70         80         90         100        110       ISO 
VOLTS  BETWEEN  TERMINALS  AND  NEUTRAL  POINT 

2U  Fig.  298  a 


658 


ALTERNATING   CURRENTS. 


Figure  297  gives  the  curves  of  efficiency,  power 
factor,  and  current  of  an  Oerlikon  100  H.P.  three-phase 
motor  of  eighteen  poles,  working  under  a  pressure  of 
1 730  volts  at  a  frequency  of  50  and  a  speed  of  320  revo- 


6000} 


aoool 


030001 


2000 


1450  2 

tr 
u 


353 


.00 


cc 

UJ 

„! 

Q 
50 


1000 


2000 


3000  1000 

WATTS  OUTPUT 

Fig.  298  b 


5000 


lutions  per  minute.  Figure  298  a,  b,  c  gives  similar 
curves  for  a  3  H.P.  three-phase  Oerlikon  motor  running 
at  a  frequency  of  50  and  a  speed  of  1500.  A  Tesla 
50  H.P.  two-phase  motor  running  at  a  frequency  of 
25  and  a  speed  of  750  revolutions  per  minute  gives  an 


ALTERNATING-CURRENT   MOTORS. 


659 


CURRENT  IN  AMPERES  AND  TORQUE  IN  KILOGRAMS  AT  8  CM.  RADIUS 
Fig.    298  C 


4800 


4200 


HORSE   POWER 

Fig.  299  a 


66o 


ALTERNATING   CURRENTS. 


efficiency  of  89  per  cent  and  a  slip  of  2  per  cent  at  full 
load  and  a  maximum  efficiency  of  91  per  cent  at  about 
three-fourths  load.  Figure  302  gives  the  curves  of  effi- 


1350 


1060 


75 


30 


15 


X 


U4 


HORSE   POWER 

Pig.  299  b 


18 


15 


1^ 


ciency  of  a  single-phase  2  H.P.  motor,  running  at  fre- 
quencies of  42  and  50.  Figures  299*2  and  299  £  give 
the  curves  of  efficiency,  current  per  phase,  apparent 
watts,  true  watts,  and  regulation  for  a  3  H.P.  Brown 


ALTERNATING-CURRENT   MOTORS. 


66 1 


two-phase  motor,  running  at  a  frequency  of  40  and  a 
speed  of  1200  when  operated  on  two  phases  and  when 
operated  on  one  phase.* 

In  Fig.  300,  curve  A  shows  the  relation  between  cur- 
rent and  starting  torque  of  a  three-phase  motor  when  a 
proper  resistance  is  introduced  into  the  armature  circuit. 
Curve  B  shows  the  same  for  the  motor  without  external 
armature  resistance ;  this  would  give  the  relations  ob- 


240 
220 
200 
180 

160 

h 

5  i«) 

e 
§120 
U 
100 

80 

60 
10 

ao 

0 

^ 

^ 

x 

J 

/ 

? 

A 

/ 

/J 

/ 

ji 

/  , 

/ 

/ 

• 

/ 

/ 

/ 

/ 

/ 

/ 

0     20     40      60     80    100    120   140  160    180    200 
TORQUE 

Fig-.  3OO 

taining  if  the  resistance  were  placed  in  the  field  circuit. 
Curve  C  gives  the  same  relations  for  a  15  H.P.  motor, 
which  has  a  full-load  torque  of  52.5  Ibs. 

184.  Weight  Efficiency.  —  The  following  table  gives 
the  weights  per  horse-power  of  three  types  of  American 
induction  motors. 


Boucherot,  Bull.  .S0<r.  Inter.  <&.$•  Jzlectriciens,  Vol.  II.,  p.  264. 


662 


ALTERNATING   CURRENTS. 


Horse- 
power. 

TYPE  NUMBER  i. 

TYPE  NUMBER  2. 

TYPE  NUMBER  3. 

Revolutions 
per  minute. 

Pounds 
per  H.P. 

Revolutions 
per  minute. 

Pounds 
per  H.P. 

Revolutions 
per  minute. 

Pounds 
per  H.P. 

I 

2666 

246 

I800 

'5° 

1800 

275 

2 

2666 

I65 

— 

— 

I800 

'75 

3 

2000 

1  80 

I800 

I67 

— 

— 

5 

2000 

164 

1200 

I40 

1200 

i*5 

7 

2OOO 

182 

— 

— 

— 

— 

7* 

— 

— 

I2OO 

1  60 

— 

— 

10 

I6OO 

125 

I2OO 

1  2O 

I2OO 

85 

IS 

I6OO 

IOI 

I2OO 

IOO 

900 

80 

20 

1333 

"05 

I2OO 

75 

900 

IOO 

30 

1333 

97 

1200 

67 

900 

88 

40 

1333 

90 

1200 

62 

— 

— 

50 

1333 

84 

900 

84 

720 

IOO 

Speeds  given  are  field  speeds  or  "  theoretical  speeds," 
and  the  full-load  speeds  are  less  by  from  2  to  10  per 
cent.  Frequencies  to  which  type  number  I  is  adapted 
are -i  33  and  66|.  The  other  types  are  adapted  to  a 
frequency  of  60. 

S.  P.  Thompson  gives  the  following  data  as  repre- 
senting European  induction  motors.* 


Horse-power. 

Weight,  pounds 
per  horse-power. 

Horse-power. 

Weight,  pounds 
per  horse-power. 

2 

120 

50 

7° 

6 

IOO 

70 

66 

13 

88 

IOO 

58 

Figure    301    gives   the    averages   for   the    American 
motors  in  the  form  of  a  curve  (A),  with  the  addition  of 


*  Thompson's  Polyphase  Electric  Currents,  p.  209. 


ALTERNATING-CURRENT   MOTORS. 


663 


the  curve  (B)  for  continuous-current  machines  repro- 
duced from  the  equations  on  page  263  of  Vol.  I.  This 
shows  that  the  alternating-current  machines  are  the 
lighter  in  the  larger  sizes,  on  account  of  their  higher 
speeds,  but  for  equal  speeds  the  two  classes  of  machines 
are  very  near  an  equality  in  respect  to  weight  efficiency. 


1 

I 

\ 

- 

1 

u 

\\ 

V 

* 

s^ 

§ 

^^, 

i 

"S, 

^ 

•  —  1 

•^ 

•^~ 

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—  ^. 

—  . 

= 

= 

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MM 

EMM. 

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VE 

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0 

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0 

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I 

i 

i 

£ 

0 

0 

7 

Fig.  301 

185.  Effect  of  Frequency.*  An  examination  of  the 
formulas  relating  to  the  design  of  induction  motors 
shows  that  the  frequency  of  the  current  for  which  a 
machine  is  designed  does  not  affect  its  efficiency,  slip, 
power  factor,  or  starting  torque,  but  that  for  a  given 
speed  the  number  of  poles  must  be  directly  as  the  fre- 
quency. Increasing  the  number  of  poles  of  a  given 

*  Steinmetz,  Trans.  Amer.  Inst.  E.  E.,  Vol.  1 1,  p.  47  ;  Stanley,  Effect 
of  Frequency  in  Induction  Motors,  Electrical  Engineer,  Vol.  17,  p.  125. 


664 


ALTERNATING    CURRENTS. 


machine  reduces  the  cross-section  of  each  pole,  but  the 
number  of  lines  of  force  at  each  pole  is  equally  reduced 
so  that  the  magnetizing  current  is  unaltered.  Conse- 
quently, induction  motors  of  equal  merit  may  be 
designed  for  all  reasonable  frequencies,  though  mag- 
netic leakage  may  interfere  with  the  operation  when  the 
poles  become  too  numerous  (compare  Transformers, 
Sect.  121). 


250 


600  250  1000 

WATTS  OUTPUT 

Fig.  302 


1250 


1500 


On  the  other  hand,  when  a  machine  which  has 
been  designed  for  a  certain  frequency  is  operated  at 
another  frequency,  the  speed  is  changed  in  direct 
proportion  to  the  frequency,  its  percentage  slip  is 
practically  unaltered,  the  starting  torque  varies  in- 
versely with  the  frequency,  and  the  efficiency  and 
power  factor  both  vary  directly  with  the  frequency 
because  the  magnetic  density  is  inversely  as  the  fre- 
quency, as  in  transformers.  Figure  302  shows  the 


ALTERNATING-CURRENT   MOTORS. 


665 


curves  of  efficiency  of  an  Oerlikon  single-phase  induc- 
tion motor  for  two  frequencies ;  *  and  Figs.  303  and  304 
give  the  curves  of  efficiency  and  power  factor  for  a  50- 
frequency  Allgemeine  three-phase  motor  when  operated 
on  four  different  frequencies,  f 

The  frequencies  which  are  commonly  used  with  in- 
duction motors  cover  a  wide  range.      In   Europe,  50 


^ 

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fe= 

™  = 

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2 

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35 
40 

FREG 

UENC 

Y  





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58 

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L 

400              800              1200             1600             2000             2400             2800    30 

WATTS  OUTPUT 

Fig.  303 

seems  to  be  quite  universally  adopted  for  three-phase 
and  single-phase  motors;  while  in  this  country  the 
General  Electric  Company  prefers  60,  which  is  equally 
suitable  for  lighting  and  power  purposes.  The  West- 
inghouse  Company  has  built  two-phase  machines  for 

*  Kolben,  Design  of  Alternating-Current  Motors,  Electrical  World, 
Vol.  22,  p.  284;  and  London  Electrician,  Vol.  31,  p.  590. 

t  Jackson,  Tests  of  a  Polyphase  Motor,  Electrical  Journal,  Vol.  i, 
p.  101. 


666 


ALTERNATING  CURRENTS. 


60  frequency,  but  appears  to  prefer  about  25,  which  is 
that  adopted  at  Niagara  Falls,  for  plants  where  power 
service  is  of  greater  importance  than  the  lighting  ser- 
vice. The  Stanley  Company,  on  the  other  hand,  pre- 
fers a  frequency  of  133  for  its  two-phasers,  though  it 
also  builds  standard  machines  for  a  frequency  of  66|, 
both  of  which  are  excellent  frequencies  for  lighting 


90 

^_ 

T9~ 

70 
60 
50 
40 
30 
20 
10 

J, 

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x* 

s 

-  — 

--u 

-^ 

^x 

a 

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, 

•** 

^i 

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kl 

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*-. 

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X      " 

jl       " 



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EQUENCY 

50  60 

Fig-.  3O4 


circuits.  The  Fort  Wayne  Electric  Corporation  and 
the  Emerson  Electric  Company  build  self-starting  syn- 
chronous single-phasers  for  frequencies  of  60,  125,  and 

133- 

186.  Other  Forms  of  Induction  Motors. — The  effects 
of  a  rotary  field,  electromagnetic  repulsion,  or  magnetic 
screening  may  be  utilised  in  an  almost  indefinite  num- 
ber of  ways,  of  which  only  a  few  will  be  indicated  here. 


ALTERNATING-CURRENT   MOTORS.  667 

I .  Motor  of  Stanley  Electric  Manufacturing  Company* 
This  is  a  two-phase  machine  which  may  be  classified 
either  as  a  rotary-field  motor,  or  as  a  sort  of  double 
single-phaser ;  and  its  design  may  be  worked  out  upon 
either  hypothesis,  though  it  should  properly  be  worked 
out  upon  the  rotary-field  basis.  It  consists  essentially 
of  two  fields  set  side  by  side.  The  field  windings  are 
placed  upon  salient  poles,  and  respectively  connected 
to  the  two  circuits  of  a  two-phase  supply  system,  and 
corresponding  to  the  two  fields  are  two  armature  cores 
on  the  same  shaft,  which  carry  a  single  set  of  short- 
circuited  armature  coils.  The  two  fields  have  an  equal 
number  of  poles,  but  they  are  set  with  the  angular  posi- 
tion of  the  poles  ninety  electrical  degrees  apart,  while 
the  armature  conductors  are  laid  in  slots  straight  across 
both  armature  cores.  A  diagrammatic  development  of 
a  six-pole  machine  which  shows  the  arrangement  very 
plainly  is  given  in  Fig.  305.  A  view  of  an  armature 
is  shown  in  Fig.  306.  The  operation  of  the  machine  is 
easily  understood  by  referring  to  Fig.  305.  When  the 
armature  is  stationary  in  the  position  shown,  current  is 
induced  in  the  a  armature  windings  by  the  lower  crown 
of  poles  belonging  to  the  A  field.  The  conductors  of 
the  a  coil  lie  directly  under  the  upper  or  B  field,  and 
a  torque  tending  to  move  the  coil  is  developed,  which, 
at  any  instant,  is  proportional  to  the  product  of  the 
instantaneous  strength  of  the  B  field  and  the  current 
induced  in  the  coil  by  the  A  field.  At  the  same  time, 
the  B  field  induces  a  current  in  the  b  coil,  which  causes 

*  Electrical  World,  Vol.  21,  p.  326;  Electrical  Engineer,  Vol.  17, 
P-  5°5- 


668 


ALTERNATING   CURRENTS. 


a  torque  with  the  A  field.     The  motor  is  therefore  self- 
starting.     After  the  machine  has  rotated  through  an 


b       a 


Fig.  305 


angle  corresponding  to  one-fourth  the  polar  pitch,  both 
coils  are  inductively  acted  upon  to  an  equal  extent  by 
both  poles,  and  opposite  torques  are  produced  by  the 


Fig.  3O6 


two    fields.      A   further   rotation    places   the  b  coil    in 
inductive  relation  with  the  A  field,  and  the  a  coil  with 


ALTERNATING-CURRENT   MOTORS.  669 

the  B  field,  and  the  torque  again  becomes  positive.  It 
is  possible,  with  such  a  winding,  to  have  dead  points, 
though  they  may  be  avoided  by  a  proper  disposition  of 
the  armature  winding.  The  Stanley  motors  are  usually 
equipped  with  a  well-proportioned  external  starting  re- 
sistance which  is  introduced  into  the  armature  circuit  by 
means  of  collector  rings,  and  the  motors  have  a  fairly 
large  starting  torque.  A  set  of  short-circuited  windings, 
m  m  m,  is  placed  in  the  pole  faces  to  decrease  the 
apparent  self-inductance  of  the  armature  windings, 
and  condensers  are  used  in  parallel  with  the  field 
circuits  to  supply  the  wattless  magnetizing  current,  and 
thus  increase  the  power  factor. 

Capacity  of  Condenser  required  to  supply  Wattless  Cur- 
rent. As  the  current  of  a  condenser  is  equal  to  2  irfsE, 
a  high  frequency  and  a  high  pressure  both  serve  to 
reduce  the  capacity  of  a  condenser  which  is  required 
to  give  any  desired  current,  and  in  order  that  the  con- 
densers which  are  required  for  the  Stanley  Company's 
motors  may  be  of  reasonable  capacity,  the  motors  are 
designed  for  500  volts  pressure,  and  the  use  of  high 
frequencies  is  recommended  by  the  company.  In  illus- 
tration of  this,  supposing  that  a' 500  volt  motor  for  120 
frequency  requires  20  microfarads  to  exactly  supply  its 
wattless  current,  an  exactly  similar  machine  designed 
for  a  frequency  of  60  would  require  from  40  to  50 
microfarads,  while  if  the  pressure  is  also  reduced  to 
250  volts  the  capacity  required  is  increased  to  from  160 
to  250  microfarads.  Figure  307  shows  the  regulation, 
power  factor,  and  efficiency  as  a  function  of  load  for  a 
2  H.P.  Stanley  motor,  and  Fig.  308  shows  the  same  for 


670 


ALTERNATING    CURRENTS. 


a  6  H.P.  machine.  The  armature  core  losses  in  Stanley 
machines  are  similar  to  those  of  single-phasers,  and  are, 
therefore,  greater  than  those  of  plain  rotary-field  ma- 
chines. A  properly  built  machine  of  this  type  gives  a 
true  rotary  field  in  its  effect  on  the  armature  windings, 


z  .  _  .  I 

52500  !---Ern=r=--- 
cc 

U  

£i;-::::::::::::::::: 

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4000  

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50 

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<  !  1  1  ?  i 

---  30 
~-\  20 

-  10 

OUTPUT  IN  HORSE  POWER   . 

Fig.  307 

as  was  early  pointed  out  by  Sahulka,*  but  the  effect  on 
the  armature  core  losses  is  similar  to  that  found  in  a 
single-phase  machine.  The  number  of  pairs  of  poles 
in  the  rotating  field  is  equal  to  the  number  of  salient 
poles  on  one  ring. 

2.    Shallenberger  Meter.      The  running  parts  of  the 

*  .Sahulka's   Ueber  Wechselstrom-Motoren  mit  magnetischem  Drehfelde. 


ALTERNATING-CURRENT   MOTORS. 


6/1 


electrical  meter  manufactured  by  the  Westinghouse 
Company  consist  essentially  of  a  single-phase  induction 
motor.  The  armature  consists  of  an  iron  disc,  across 
one  diameter  of  which  is  wound  a  stationary  coil  that 
carries  the  main  current.  A  short-circuited  coil,  consist- 
ing of  heavy  copper  strips,  lies  within  and  at  a  slight 
angle  with  the  main  coil,  and  the  lagging  induced  cur- 


2.0  3.0  4.0 

OUTPUT   IN    HORSE   POWER 


Fig.  308 

rent  in  this  coil  sets  up  a  magnetic  field  which  joins 
with  the  magnetism  of  the  main  coil  to  set  up  an  irreg- 
ularly rotating  field  with  a  frequency  equal  to  that  of 
the  main  current.  This  causes  the  iron  armature  to 
rotate.  The  strength  of  the  resultant  field  and  the 
torque  on  the  armature  are  proportional  to  the  main  cur- 
rent. A  proper  retarding  force  is  used  to  cause  the 
armature  to  rotate  at  a  speed  directly  proportional  to 
the  main  current  as  long  as  the  frequency  is  constant. 


6/2  ALTERNATING   CURRENTS. 

The  speed  of  the  armature  depends  on  the  frequency 
of  the  main  current. 

3.  Scheeffer  Meter.     The  running  parts  of   a  record- 
ing wattmeter,  manufactured  by  the  Diamond  Electric 
Company,    consist    of    a    split-phase    induction    motor. 
The  armature  is  an  iron  cylinder,  which  is  embraced 
by   a   three-legged   magnet    made   of    iron    stampings, 
upon   one  leg  of  which  is  wound  a  coil   carrying   the 
main  current,  and  upon  another  leg  is  wound  a  shunt 
or  pressure  coil.     The  magnetism  set  up  by  these  two 
coils,  in  which  the  current  has  different  phases,  sets  up 
an  irregular  rotary  field,  the  strength  of  which  depends 
upon  the  product  of  the  main  current  and  pressure,  by 
means  of  which  the  armature  is  driven.     By  a  proper 
retarding  force  the  armature  may  be  caused  to  run  at 
a  speed  which  is  directly  proportional  to  the  watts  in 
the  circuit.     It  is  possible  to  adjust  the  magnetic  den- 
sity in  the  cores,  by  adjusting  the  resistance  of  the  shunt 
coil,  so  that  the  speed  of  the  armature  is  practically  in- 
dependent of   the  frequency  within  the  ordinary  com- 
mercial limits. 

4.  Ferranti  Meter.     The  armature  in  this  is  an  iron 
disc  which  is  embraced  by  two  elongated  pole  pieces, 
and  these  are  surrounded  at  equal  intervals  by  short-cir- 
cuited copper  bands.     The   bands   exert  what  may  be 
called  a  shielding  effect  on  the  magnetism,  and  cause 
magnetic   poles   to   apparently   creep   along   the   pole- 
pieces.     These  cause   the   revolution  of  the  armature. 
The  speed  of  the  poles  and  the  armature  torque  depend 
upon  the  strength  of  the  magnetism  set  up  in  the  main 
coil 


ALTERNATING-CURRENT   MOTORS.  6/3 

Thomson's  recording  wattmeter  consists  of  a  small 
motor  with  a  Gramme  armature,  and  without  iron  in 
fields  or  armature  cores,  so  that  it  works  equally  well 
on  alternating  or  continuous  currents.  It  is  not  an 
induction  motor,  but  seems  to  require  notice  amongst 
the  other  meters  already  described.  It  is  well  known 
that  a  small  series  motor,  with  either  Gramme  or  Sie- 
mens armature,  will  run  on  alternating  circuits  exactly 
as  it  runs  on  continuous-current  circuits,  but  its  power 
factor  is  so  minute  as  to  preclude  its  commercial  value. 
In  the  Thomson  meter,  the  main  current  passes  through 
the  field  coils,  and  the  armature  is  connected  directly 
across  the  leads  through  a  large  non-inductive  resist- 
ance. 

The  only  other  type  of  induction  motor  to  which 
attention  can  be  given  is  that  developed  by  Mr.  C.  P. 
Steinmetz,  together  with  a  special  arrangement  of  alter- 
nator windings  and  transmission  lines  which  is  called 
the  Monocyclic  System. 

187.  Monocyclic  System.*  —  A  diagram  of  a  "mono- 
cyclic  "  alternator  armature  is  shown  in  Fig.  309,  and 
the  same  is  developed  in  Fig.  310^.  The  winding 
on  this  alternator  consists  of  an  ordinary  coil  winding 
in  slots  or  grooves,  which  may  be  called  the  main 
winding  or  coil,  and  an  auxiliary  or  "  teaser  "  winding 
placed  in  smaller  slots  half-way  between  the  main 
slots.  The  electrical  pressure  developed  in  the  auxil- 
iary winding  has,  on  account  of  the  position  of  the 
coil,  a  phase  difference  of  ninety  degrees  from  that 
developed  in  the  main  winding,  exactly  as  would 

*  Electrical  World,  Vol.  25,  pp.  182  and  302. 
2  x 


6/4 


ALTERNATING   CURRENTS. 


be  the  case  in  a  two-phase  alternator,  but  one  end 
of  the  auxiliary  coil  is  connected  to  the  middle  of 
the  main  coil.  The  free  end  of  the  auxiliary  coil  and 


Fig.  3O9 


Fig-.  31O  a 


the  ends  of  the  main  coil  are  connected  to  separate 
collector  rings.  If  the  number  of  turns  in  the  auxiliary 
coil  bore  the  relation  to  the  number  in  the  main  coil  of 

— _:  i,  the  pressure  measured  between  the  collector 
rings  taken  in  pairs  would  be  equal  for  each  pair,  and 


A                   p 

o 

7 

\ 

/ 

\ 
\ 

/               A                    D                    B 

\ 

/                           ^                     1                     ?* 

\ 

/                                                               ^-s,                                                          x<--x*' 

\ 

^                                                                                                              **  N.                                               x''*' 

N 

/                                                                                                                                 ^»              1            -X''X 

\ 

/                                                                                                                                             ^^^ 

\ 
\ 

C 

\ 

/ 

C 

Fig.  31O  b                                            Fig.  31O  c 

the  machine  would  be  a  balanced  three-phase  generator 
giving  three  equal  pressures  at  120°  difference  of  phase 
(Fig.  310$).  But  in  the  monocyclic  generator  the  aux- 


ALTERNATING-CURRENT   MOTORS. 


675 


iliary  coil  has  only  one-fourth  as  many  turns  as  the  main 
coil,  and  therefore  the  three  phases  developed  by  the 
machine  are  not  120°  apart,  but  have  the  angular  rela- 
tions shown  in  Fig.  310*:,  in  which  AB  is  the  pressure 
measured  between  the  main  terminals,  AC  and  BC  press- 
ures measured  between  the  auxiliary  terminal  and  the 
main  terminals,  and  CD  the  pressure  developed  in  the 
auxiliary  coil.  The  angle  A  CD  is  nearly  60°  and  A  C  is 
nearly  .56  of  AB. 

If  two  transformers  are  connected  in  circuit  with  a 
monocyclic  generator,  it  is  possible  to  get  a  three-phase 


SECONDARY 


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". 

PR 

MARY 

Fig.  311  a                            Fig.  311  b 

secondary  circuit  with  120°  difference  of  phase  by  the 
arrangement  shown  in  Fig.  3 1 1  a,  provided  the  ratios  of 
the  number  of  turns  cb  to  ab  and  CB  to  A  B  are  properly 
proportioned.  Two  ordinary  transformers  with  the  same 
ratio  of  transformation  may  be  used  as  shown  in  Fig. 
3 1 1  £,  one  of  the  secondaries  being  reversed,  though  the 
pressures  in  the  three  circuits  are  then  not  exactly  equal. 
The  way  in  which  the  pressures  come  out  in  this  case 


6;6 


ALTERNATING   CURRENTS. 


is  illustrated  in  Fig.  3 1 1  c,  where  A  C  is  the  pressure 
measured  from  a  to  c,  £Cis  the  pressure  measured  from 
b  to  c,  Bf  C  shows  the  phase  of  the  pressure  BC  without 
a  reversed  transformer,  and  AB  is  the  resultant  press- 


Fig.  311  c 

ure  measured  between  a  and  b.  The  last  is  equal  to 
twice  the  pressure  developed  in  the  teaser  coil,  or  CD 
in  Fig.  310^.  This  may  be  treated  as  a  regular  three- 
phase  system  in  bad  balance,  but  the  system  was  de- 
signed to  be  operated  with  a  special  two-coil  induction 
motor  which  is  shown  diagrammatically  in  Fig.  312. 
The  field  of  this  motor  is  wound  with  two  coils,  one  of 
which,  mt  is  connected  to  the  main  circuit,  and  the  other, 


Fig.  312 

mf ,  which  has  fewer  turns,  is  connected  to  the  auxiliary 
conductor.  The  motor  acts  in  starting  as  an  unbalanced 
three-phaser,  but  after  getting  under  way  takes  most  of 
its  power  from  the  main  circuit.  This  arrangement  was 


ALTERNATING-CURRENT   MOTORS.  677 

intended  to  avoid  the  unbalancing  which  is  likely  to 
occur  in  polyphase  systems  where  lighting  and  power 
are  used  together.  The  monocyclic  system  is  essentially 
a  single-phase  system ;  all  the  lighting  apparatus  is 
operated  from  the  main  circuit,  and  while  the  motors 
are,  broadly  speaking,  polyphasers,  and  may  be  ordinary 
three-phasers,  they  operate  more  after  the  manner  of 
single-phasers  which  are  started  by  splitting  the  phase, 
than  as  balanced  three-phasers.  Quite  a  large  number 
of  monocyclic  alternators  have  been  put  in  service  in 
this  country  as  ordinary  single-phase  lighting  genera- 
tors *  but  comparatively  few  have  been  put  into  operation 
for  use  in  a  combined  light  and  power  service.  The 
motors  in  use,  where  the  combined  service  is  furnished, 
are  standard  three-phasers. 

The  accomplished  designer  of  the  monocyclic  system 
has  made  many  plans  in  which  it  is  proposed  to  utilize 
in  remarkably  varied  ways  the  flexibility  of  alternating- 
current  machinery  of  the  induction  types.  One  of  these 
plans  is  shown  diagrammatically  in  Fig.  3 1 3,  and  is  in- 
troduced here  for  the  purpose  of  illustrating  more  fully 
the  varied  purposes  to  which  alternating-current  appa- 
ratus lends  itself,  f  G,  in  the  figure,  is  an  ordinary 
single-phase  generator,  with  its  field  excited  by  exciter, 
£,  and  its  armature,  A,  connected,  through  the  collector 
rings,  rr,  to  feeders,  to  which  lighting  circuits  may  be 
directly  connected  through  transformers,  as  at  a.  At  b 


*  Jackson  and  Fortenbaugh,  Some  Observations  on  a  300  K.  W. 
Monocyclic  Alternator,  Trans.  Amer.  Inst.  E.  E.,  Vol.  12,  p.  350. 

t  See  also  Emmett,  Existing  Commercial  Applications  of  Electrical 
Power  from  Niagara  Falls,  Trans.  Amer.  Inst.  E.  E.,  Vol.  12,  p.  482. 


6;8 


ALTERNATING   CURRENTS. 


is  a  circuit  containing  several  monocyclic  motors,  which, 
after  one  is  started,  furnish  each  other  the  current 
required  for  the  auxiliary  winding;  while  at  M  is  a 
synchronous  motor  wound  like  a  monocyclic  generator, 
but  which  operates  as  a  single-phase  synchronous  motor 


Fig.  313 

with  its  main  coil  connected  to  the  generator  circuit. 
Its  auxiliary  coil  serves  to  furnish  the  additional  current 
needed  to  operate  plain  three-phase  induction  motors, 
7,  7,  which  are  on  the  circuit. 

188.  Effect  of  the  Form  of  Curves  of  Pressure.  —  The 
effect  of  distorted  curves  upon  the  operation  of  induc- 
tion motors  depends  upon  the  number  of  phases.  The 
harmonics  of  three  and  five  times  the  fundamental  fre- 
quency are  the  only  ones  which  need  be  considered ; 
and  indeed,  that  of  three  times  the  frequency  is  the 
only  one  which  has  an  appreciable  influence  (Sect.  30, 
and  Appendix  A).  In  single-phase  motors  the  har- 


ALTERNATING-CURRENT   MOTORS.  6/9 

monies  must  affect  the  magnetic  field  exactly  as  they 
affect  that  of  a  transformer,  so  that  peaked  pressure 
curves  should  cause  a  decrease  in  core  losses,  and  the 
operation  of  the  motor  should  not  be  otherwise  greatly 
influenced.  In  polyphase  motors,  however,  the  har- 
monics may  set  up  a  rotating  field  of  their  own,  which 
is  superposed  upon  the  regular  field,  and  may  interfere 
with  the  operation  of  the  machine.  The  harmonics  with 
three  times  the  fundamental  frequency  belonging  to 
the  two  circuits  of  a  two-phase  system,  have  a  phase 
difference  of  90°  (Fig.  314),  and  these  set  up  a  super- 
posed rotating  field  in  the  induction  motor  which  has 
a  field  velocity  of  three  times  that  of  the  main  field. 
The  figure  shows  that  the  harmonics  of  triple  frequency, 
belonging  to  the  two  phases,  are  reversed  in  relative 
position  compared  with  the  fundamental  waves.  The 
field  due  to  these  harmonics  rotates  in  the  reverse 
direction  from  that  of  the  main  field,  and  therefore 
tends  directly  to  decrease  the  torque  of  the  motor  and 
to  increase  the  slip.  The  field  due  to  the  harmonics 
of  five  times  the  frequency  rotates  in  the  same  direc- 
tion as  the  main  field,  and  its  only  disadvantageous 
effect  is  in  causing  eddy  currents  which  may  slightly 
decrease  the  efficiency  of  the  motor.  The  frequency  of 
the  harmonic  curves  is  indicated  in  the  figure  by  sub- 
scripts. 

In  three-phase  circuits  the  harmonics  of  triple  fre- 
quency belonging  to  the  different  currents  are  directly 
superposed  in  phase  as  is  shown  in  Fig.  315,  and 
therefore  the  superposed  field  which  they  cause  in 
three-phase  induction  motors  is  a  stationary  one  whose 


68o 


ALTERNATING   CURRENTS. 


\ 


\ 


\ 


ALTERNATING-CURRENT   MOTORS.  68 1 


o 


o 


682         ALTERNATING  CURRENTS. 

influence  is  only  to  decrease  the  efficiency  by  setting 
up  extra  core  losses.  The  figure  also  shows  that  the 
harmonics  of  five  frequencies  have  120°  difference  of 
phase  and  are  in  reversed  order,  so  that  they  set  up  a 
reverse  rotating  field,  and  if  they  are  in  much  strength 
may  affect  the  torque. 

188  a.  Reversing  Polyphase  Motors.  —  Polyphase  mo- 
tors may  be  reversed  by  reversing  the  direction  of  rota- 
tion of  the  field. 

In  two-phase  motors  with  independent  circuits,  re- 
versing the  terminal  connections  of  either  circuit  will 
effect  the  reversal  of  rotation,  but  reversing  the  ter- 
minals of  both  circuits  will  not  alter  the  direction  of 
rotation.  Two-phase  motors  with  three-wire  connec- 
tions cannot  be  reversed  by  any  change  of  the  external 
connections. 

The  direction  of  rotation  of  three-phase  motors  may 
be  reversed  by  interchanging  the  connections  of  any 
pair  of  leads. 


POLYPHASE   TRANSFORMERS.  683 


CHAPTER   XV. 

POLYPHASE    TRANSFORMERS. 

189.    Stationary  Transformers  for  Polyphase  Circuits. 

-The  transformation  of  pressure  in  polyphase  circuits 
may  be  compassed  by  using  single-phase  transformers 
in  groups.  A  two-phase  circuit  then  requires  two  trans- 
formers at  each  point  of  transformation,  and  a  three- 
phase  circuit  requires  either  two  or  three  transformers. 
The  individual  transformers  must  each  have  a  capacity 
equal  to  the  power  required  to  be  transformed  in  each 
phase  divided  by  the  power  factor  of  the  secondary  cir- 
cuit. As  the  power  factor  of  an  incandescent  lamp 
circuit  is  practically  100,  and  as  circuits  supplying 
motors  are  likely  to  have  a  full-load  power  factor  at 
best  as  small  as  80,  it  is  evident  that  transformers  which 
supply  currents  to  motors  must  be  of  greater  capacity 
than  those  which  supply  an  equal  power  to  incan- 
descent lamps.  This  rule  applies  equally  to  single-phase 
and  polyphase  circuits  and  is  important  to  bear  in  mind 
at  the  present  time  when  alternating-current  motors  are 
coming  into  use.  Figure  316  shows  the  connections 
of  a  three-phase  circuit  with  two  and  with  three  trans- 
formers. 

A  saving  in  the  amount  of  material  used,  and  there- 
fore also  in  the  economy  of  operation,  may  be  effected 


684 


ALTERNATING  CURRENTS. 


by  combining  the  magnetic  circuits  of  the  individual 
transformers  in  the  several  phases,  exactly  as  polyphase 
electric  circuits  are  combined  into  common  wires  (Chap. 
XIII.).  Figure  317  represents  a  two-phase  transformer 
with  a  combined  magnetic  circuit.  Since  the  phases  of 
the  magnetism  in  the  two  halves  of  the  transformer  are 
90°  apart,  the  resultant  magnetism  in  the  middle  tongue 


I L 


3  J*  vvwv?~l>vwvwK/s/wJ  | 


3D  HI 


00        O 


IvwytW/wJ  I/Wv*TkAA/vJ 


Fig.  316 

is  A/2  times  as  great  as  that  in  the  cores  under  the 
windings,  so  that  this  central  tongue  must  have  A/2 
times  as  great  a  cross-section  as  the  remainder  of  the 
magnetic  circuit.  There  is  a  saving  of  iron  in  the  com- 
bined transformer,  as  compared  with  two  independent 
transformers,  which  is  equal  to  ~\/2  times  the  weight 
of  the  central  tongue.  This  is  of  little  moment  in  small 
transformers,  but  may  make  quite  a  difference  in  the 


POLYPHASE   TRANSFORMERS. 


685 


cost  and  efficiency  of  transformers  of  very  large  capac- 
ity.    The  same  sort  of  combination  may  be  effected  in 


Fig.  317 


three-phase  transformers,  and  the  magnetic  circuit  may 
be  coupled  in  either  the  star  or  the  mesh  arrangement. 


Fig.  318 

Figure  318  shows  a  three-phase  transformer  used  by 
Siemens  and  Halske,  and  other s,  in  which  the  magnetism 


686  ALTERNATING   CURRENTS. 

in  the  yokes,  DD\  which  join  the  cores  A,  B,  and  C,  is 
V3  times  as  great  as  that  in  the  cores,  and  the  con- 
struction allows  a  considerable  economy  in  comparison 
with  separate  transformers  in  the  three  phases  ;  while,  if 
the  windings  for  the  three  phases  are  placed  on  the  three 
sides  of  a  triangle  or  are  arranged  in  consecutive  order 
on  a  ring,  the  core  must  be  V3  times  as  great  as  would 
be  required  for  one  phase  alone  and  the  saving  of  iron 
is  in  the  relation  of  3  :  V^. 

After  giving  due  regard  to  the  resultant  magnetism 
in  the  cores,  the  principles  and  practice  in  transformer 
design,  construction,  and  testing,  which  have  already 
been  fully  developed  (Chaps.  X.,  XL,  and  XII.),  are 
directly  applicable  to  polyphase  transformers. 

190.  Transformation  of  Phases.  —  Arrangements  for 
transforming  one  polyphase  system  into  another  sys- 
tem with  a  different  number  of  phases  may  be  readily 
developed  from  the  principles  which  have  been  fully  set 
forth  in  the  chapters  on  single-phase  transformers  and  in- 
duction motors.  Quite  a  number  of  commercial  devices 
for  this  purpose  have  been  proposed.  Mr.  C.  F.  Scott  * 
has  patented  a  method  for  transforming  two-phases  into 
three-phases  which  has  been  in  some  commercial  ser- 
vice. It  is  arranged  as  follows:  in  Fig.  319*2,  the 
primaries  of  the  transformers  M  and  M1  are  connected 
to  a  two-phase  source.  The  secondary  of  M  is  attached 
to  the  middle  of  the  secondary  of  M' ,  as  shown  at  O. 

The  secondary  of  M  has  — -  times  the  turns  of  that  of 

2 

*  Polyphase   Transmission,  Electrical   World,  Vol.  23,   p.   358;    Lond. 
Electrician,  Vol.  32,  p.  640. 


POLYPHASE   TRANSFORMERS. 


68; 


M'.  Then  in  Fig.  319^  the  line  OB  represents  the 
pressure  between  the  points  O  and  B  in  the  former 
figure,  OC  that  between  O  and  C,  and  OA  that  between 
O  and  A.  OA  must  be  at  right  angles  to  OB  or  OC,  as 
the  two-phases  of  the  primaries  are  90°  apart.  Thus 
it  is  seen  that  between  the  points  A,  B,  and  C  three 
equal  pressures  are  set  up  at  120°  apart.  By  reversing 
the  apparatus,  three-phases  may  be  transformed  into 
two-phases.  Other  arrangements  for  effecting  the  same 


Fig.  319  a 


COB 
Fig.  319  b 


result  may  be  readily  suggested,  such  as  that  shown  in 
Fig.  320,  where  aal r,  bb\  ccl  represent  the  three  coils 
of  a  three-phase  winding  uniformly  placed  upon  a  ring 
core,  and  AA',  BB*  a  uniform  two-phase  winding.  If 
one  of  the  windings  is  connected  to  an  appropriate 
polyphase  circuit  it  causes  a  rotary  field  to  be  set  up  in 
the  core  which  sets  up  a  polyphase  current  in  the 
circuit  of  the  other  set  of  coils.  In  this  case  the  num- 
ber of  phases  in  the  secondary  circuit  is  independent  of 
the  number  of  primary  phases  and  depends  only  upon 
the  number  and  arrangement  of  the  secondary  coils. 
The  magnetic  circuit  should  be  completed  by  filling  the 
central  space  with  iron  stampings. 

The  transformation  of   single-phase   into   polyphase 


688  ALTERNATING   CURRENTS. 

currents  by  means  of  stationary  transformers  may  be 
accomplished  by  phase-splitting  devices,*  but  no  satis- 
factory commercial  method  has  been  developed  which 
does  not  include  moving  parts  in  the  transformer. 


191.  Rotary  Transformers.  —  The  possibility  of  con- 
verting a  continuous-current  dynamo  into  a  single-phase 
alternator  was  referred  to  in  Section  5  and  later  sections, 
and  a  machine  so  constructed  with  a  continuous-current 
commutator  and  alternating-current  collector  rings  may 
be  used  to  convert  a  continuous  current  which  is  fed  into 
its  commutator  end,  and  by  which  it  is  driven,  into  an 
alternating  current  which  is  taken  from  the  collector 
rings.  Or,  the  transformation  may  be  from  alternat- 

*  Bradley,  Phasing  Transformers,  Trans.  Amer.  Inst.  E.  E.,  Vol.  12, 
p.  505 ;  Steinmetz,  Some  Features  of  Alternating-Current  Systems,  Trans. 
Amer.  Inst.  E.  E.,  Vol.  12,  p.  329. 


POLYPHASE   TRANSFORMERS. 


689 


ing  to  continuous  currents,  if  the  armature  is  prop- 
erly synchronized  so  that  it  runs  as  a  synchronous 
motor. 

It  is  possible  in  the  same  manner  to  make  a  two- 
phaser  to  be  used  with  separate  circuits  out  of  any 
continuous-current  machine  with  Gramme  or  Siemens 
armature,  by  arranging  four  collector  rings  on  the  shaft 
and  connecting  them  to  the  armature  windings  at  points 
which  are  90  electrical  degrees  apart.  It  is  also  possi- 
ble to  make  a  three-phaser  out  of  a  continuous-current 


a     b    c 


Fig-.  321 

machine  by  arranging  three  collector  rings  on  the 
shaft,  and  connecting  them  to  the  armature  winding, 
at  1 20  electrical  degrees  apart.  Such  machines  may 
be  used  to  transform  continuous  currents  into  polyphase 
currents  or  vice  versa.  (See  Fig.  321.)  In  the  case  of 
two-phasers,  it  is  evident  that  the  maximum  value  of 
the  alternating  pressure  is  equal  to  the  value  of  the 


2Y 


690  ALTERNATING   CURRENTS. 

continuous  pressure,  and  hence  the  ratio  of  transfor- 
mation is  theoretically  I  :  V2.  In  the  case  of  three- 
phasers  a  little  consideration  will  show  that  the  ratio 
of  transformation  is  I  :  V^.  These  theoretical  values 
are  found  to  hold  very  closely  in  commercial  machines. 
They  are  independent  of  the  speed  of  the  machines  and 
of  the  strength  of  the  fields,  provided  armature  reac- 
tions are  small. 

Machines  so  constructed  are  called  Rotary  Transfor- 
mers. They  will  run  in  synchronism  when  fed  with 
alternating  currents,  and  their  speed  therefore  depends 
upon  the  number  of  poles  in  the  field  and  the  frequency 
of  the  currents.  Polyphase  rotary  transformers  are 
generally  self-starting  from  the  alternating-current  end 
by  the  effect  of  foucault  currents  set  up  in  the  pole 
pieces  by  the  rotary  field  which  exists  in  the  armature 
when  it  is  not  in  synchronism.  The  starting  torque 
may  be  increased,  as  in  polyphase  synchronous  motors, 
by  embedding  copper  "  induction  bars  "  across  the  pole 
faces.  After  a  rotary  transformer  fed  by  an  alternating 
current  is  in  synchronism,  its  fields  may  be  magnetized 
by  the  continuous  current  produced  by  itself  and  col- 
lected from  its  commutator. 

In  connecting  the  armature  windings  of  rotary  trans- 
formers to  the  collector  rings,  the  relative  angles  cor- 
responding to  the  current  phases  must  be  carefully 
distinguished  (compare  Sect.  102  a).  One  complete 
revolution  of  an  armature  in  a  two-pole  field  corre- 
sponds to  one  complete  period  of  the  alternating  cur- 
rent, and  therefore  360  mechanical  degrees  corresponds 
to  360  electrical  degrees,  but  in  multipolar  machines  a 


POLYPHASE  TRANSFORMERS.        691 

rotation  of  the  armature  equal  to  twice  the  angular  pitch 
of  the  poles  corresponds  to  one  complete  period,  so  that, 
in  general,  the  relation  of  electrical  degrees  to  mechani- 
cal degrees  is/:  i,  where/  is  the  number  of  pairs  of 
poles.  Two-pole  rotary  transformers  evidently  utilize 
the  whole  of  the  armature  winding  with  each  collector 
ring  connected  to  a  single  point,  and  the  same  is  true 
of  multipolar  machines  with  series  path  windings  (Vol.  I., 
p.  276).  If  single  connections  to  the  collector  rings  are 
used  in  multipolar  machines  with  multiple  path  wind- 
ings, a  portion  only  of  the  armature,  corresponding  to 
360  electrical  degrees,  is  occupied  in  the  delivery  of 
alternating  currents,  and  the  armature  capacity  is  there- 
fore not  fully  utilized.  To  fully  utilize  the  armature  in 
this  case,  each  collector  ring  must  be  connected  to  the 
winding  at  as  many  points  as  there  are  pairs  of  poles, 
the  points  being  360  electrical  degrees  apart. 

The  capacity  of  a  rotary  transformer  of  this  type  is 
greater  than  the  same  machine  used  either  as  an  alter- 
nator or  as  a  continuous-current  generator,  and  the 
excess  capacity  increases  with  the  number  of  phases. 
This  is  due  to  the  fact  that  the  transformed  current 
does  not  traverse  all  of  the  armature  conductors,  but 
takes  the  path  from  the  continuous-current  brushes  to 
the  alternating-current  brushes  in  which  it  meets  the 
least  opposition,  and  the  heating  and  armature  reactions 
for  a  given  output  are  reduced.*  The  ratio  of  trans- 
formation of  the  machine  when  operated  to  transform 
alternating  currents  into  continuous  currents  may  be 

*  Mershon,  Output  of  Polyphase  Generators,  Electrical  World,  Vol.  25, 
p.  684. 


692  ALTERNATING   CURRENTS. 

increased  by  unbalancing  the  polyphase  circuit  by  the 
introduction  of  unequal  inductances. 

Rotary  transformers  are  also  constructed  with  two 
independent  armature  windings,  or  by  rigidly  connect- 
ing independent  machines  together. 


APPENDICES. 

A.  THE  APPLICATION  OF  FOURIER'S  THEOREM  TO  ALTERNATING- 

CURRENT  CURVES. 

B.  THE  CHARACTERISTIC  FEATURES  OF  ALTERNATING-CURRENT 

CURVES. 

C.  OSCILLATORY  DISCHARGES. 

D.  ELECTRICAL  RESONANCE. 


APPENDIX   A. 

THE  APPLICATION   OF   FOURIER'S   SERIES  TO   ALTERNATING- 
CURRENT   CURVES. 

IT  has  been  stated  in  Section  30  of  the  text  that  alternat- 
ing-current curves  may  be  represented  by  a  special  form  of 
Fourier's  series, 

e  (or  c)  =  # !  sin  a  +  a3  sin  3  a  +  a5  sin  5  a  +  etc. 
H-  b±  cosa  +  bz  cos  3 a  +  b&  cos  5  a  -f  etc., 

but  it  is  a  matter  of  some  labor  to  determine  the  constants  a 
and  b  which  apply  to  any  particular  curve.  This  may  be  done 
in  the  following  manner,  first  assuming  that  an  alternating-cur- 
rent curve  has  been  experimentally  determined  and  plotted  in 
the  usual  manner  to  rectangular  co-ordinates  and  it  is  desired 
to  find  the  constants  to  be  inserted  in  the  Fourier  series  in 
order  to  give  the  equations  of  the  curve.  Divide  the  base  of 
one  loop  of  the  curve  into  n  +  i  equal  divisions,  then  there 
will  be  n  points  between  a  =  o  and  a  =  180°,  which  will  cor- 
respond to  Aa=f^-Y,  2Aa=f2  X  l8°Y,  etc.,  and  the 

\n  +  ij  \  n  +  i  ) 

abscissa  of  any  of  the  points  may  be  represented  in  general 

by  k  Aa  =  (  )  •    Corresponding  to  each  abscissa  there  will 


,«  + 
be  an  ordinate  which  represents  a  value  of  e  (or  c),  which  may 

be  called  ek  (or  ^).    Substituting  in  the  original  equation  gives 
ck  (or  ck) 
=  «  sin +  a&  sin  3  +  ,5  sin  5  +  etc. 


695 


ALTERNATING   CURRENTS. 


696 

By  giving  k  successive  numerical  values  from  unity  to  n,  there 
are  found  n  equations  of  the  first  degree  from  which  the  values 
of  #1,  tf3,  #5,  etc.,  ^  ^3,  <£5,  etc.  to  n  terms,  may  be  determined 
by  the  usual  algebraic  methods.  Putting  m  as  a  general  sub- 
script for  a  or  b,  then 


n+i 


-f-  tn  sin  ( 


\       n+i 


nm 


n+ 


\  cosf  m- 
\    n 


V     »  +  i 

/         180 

+  <?nCOS     ^W  — 

These  expressions  may  be  written  for  convenience 


k=i         \      w  +  i/  » +  i  *=i         \ 

The  pressure  wave  of  an  alternator  is  represented  by  the 
heavy  line  of  Fig.  A,  and  the  constants  of  Fourier's  series  for 


-10, 


/?;> 


<? 


^as 


Fig.  A 

this  curve  have  been  determined  up  to  the  seventh  harmonic, 
giving  the  values 


APPENDIX   A.  697 

tf1  =  +  98.6,  ^i  =  —  i4-7> 

03  =  —  13.3,  £3  =  +18.2, 

a5  =  -     1.6,  ^5  =  —    4-8, 

07  =  +          .25,  ^7  =  +       1.2. 

The  equation  for  the  curves  as  determined  by  this  means  is 

e  =  98.6  sin  a  —  13.3  sin  3  a  —  1.6  sin  5  a  -f  .25  sin  7  a 
-  14.  7  cos  a  -f-  18.2  cos  3  a  —  4.8  cos  5  a  +  1.2  cos  7  a. 

Substituting  various  values  of  a  in  this  equation,  the  corre- 
sponding values  of  e  are  given,  and  the  corresponding  curve, 
which  is  dotted,  has  been  plotted  in  the  figure.  It  will  be 
noticed  that  the  calculated  curve  crosses  the  original  in  seven 
points,  and  very  closely  approximates  to  its  exact  form.  If  a 
larger  number  of  constants  had  been  determined,  the  calculated 
curve  would  have  crossed  the  original  curve  a  proportionally 
larger  number  of  times,  and  the  approximation  would  have  been 
still  closer.  The  number  of  times  the  calculated  curve  crosses 
the  original  curve  is  equal  to  n,  and  consequently  the  calcu- 
lated curve  cannot  exactly  coincide  with  the  original  curve 
unless  n  —  oo.  The  series  used  is  rapidly  convergent,  and  in 
this  particular  curve  the  effect  of  the  fifth  and  seventh  harmonics 
is  quite  small,  and  the  curve  is  sufficiently  well  represented  for 
practical  purposes  by  the  fundamental  and  third  harmonics,  in 
which  case  the  equation  is 

e  =  98.6  sin  a  —  13.3  sin  3  a 
—  14.7  cosa-f  18.2  cos  30,. 

The  corresponding  sine  and  cosine  terms  of  the  series 


j  +  (      03sin3a  j  +  (      ^5  sin  5  a  )  +  f  +  etc., 
+  b±  cos  a  )       1  +  bz  cos  3  a  )       1  -f  ^5  cos  5  a  j        1  +  etc. 

may  be  conceived  as  representing  the  rectangular  sine  com- 
ponents of  the  terms  of  a  single  sine  or  cosine  curve.  This  is 
illustrated  in  Fig.  B,  from  which  it  is  evident  that 


698 


ALTERNATING   CURRENTS. 


or 
Where 


am  sin  ma  ±  bm  cos  mo.  =  cm  sin  (;/za  -f-  Om) 
am  sin  wa  ±  <£TO  cos  #2a  =  cm  cos  (#/a  —  0m') . 


2,  and 


=  ^or 


Substitution  gives 
e  =  ti  sin  (a  +  Oi 

=  *-!  COS  (a  -  ^ 


cs  sin  (3  a  +  03)  +  c5  sin  (5  a  +  05)  +  etc. 

(T3  COS  (3  a  -  03')  +  ^5  COS  (5  a  -  65')  +  etc. 


Fig.  B 

The  equation  given   previously,   when  reduced   to  this  form 
(using  0),  has  the  following  constants  : 


'i=99-7, 

'3=22.5, 


=  -53°  50' 
=  +7i°  34' 


fV=    1.2,  7  =  +7834, 

and  the  equation  is 

e  =  99.7  sin  (a  —  8°  29')  —22.5  sin  (3  a  —  53°  50') 

-  5.1  sin  (5  a  +  71°  34')  +  1.2  sin  (7  a  +  78°  34'), 
and  its  value  to  a  considerable  degree  of  approximation  is 
e=  99.7  sin  (a  —  8°  29')—  22.5  sin  (3  a  —  53°  50'). 


APPENDIX   A.  699 

Following  are  examples  of  the  calculation  of  the  constants  of 
these  equations  : 


Values  of  e  from  curve  : 

<?i=i8,          <?3=7°>  ^5=125,          <?7  =  3°- 

<?2  =  42,          e^=  no,         e6=  80, 

tfj  =  Jj  18  sin  22°  -J-  -f  42  sin  45°  -f  70  sin  67°  ^  +  ••• 
+  30  sin  15  7°  £{=  98.6, 

as  =  i  j  18  sin  67°  -|-  +  42  sin  135°  -f-  70  sin  202°  |  +  • 
+  3osinii2°i|^-  13.3, 
-  +  42  cos  135°  +  70  cos  202°  ±  -\ 

+  30  COS  112°  Jj   =  l8.2. 


3 

The  effective  ordinate  of  an  alternating-current  curve  may 
be  determined  by  integrating  directly  from  its  equation.  The 
effective  value  squared  is 


7T     0 

i  C 
=—  \ 

TTc/O 

and  since 

/*7T  S*TT  f*1f 

I    sin  ;;za  sin  nada,       \    sin  ma  cos  mada,  cos  ;«a  cos 

»/o  c/o  «yo 

are  each  equal  to  zero  when  m  and  «  are  unequal  integers,  there 
results 


i    s**  a\    Cn  as2  Cn 

—  -  I    ^Va  =  —  I  _   sin2  ode.  -\  --  I    sin2  3  ada 

TTjO  7T  «/"  7T  »/0 

a?  r*  b*  /•«• 

+  V  Jo  sin2  5  a^a  +  etc-  *  vJo  cos2  ada- 

b?  r*  b?  C" 

4-  IT  I    c°s2  3  a^a  +  —  I    cos2  5  a^a  +  etc- 

7T«/0  7T*/0 


700 


ALTERNATING   CURRENTS. 


But    I    sin2  mada.   and    (    cos2  ma.da  are  always  equal   to   —  : 

Jo  Jo  2 

a?      a&2      a,2  b?      b.?      b? 

hence    E*  =  ^  +  -±  +  f  +  etc.  +-1  +  -±  +  f  +  etc., 


and 


I- 


) 


In  the  example  which  has  been  given,  the  value  of  E  calcu- 
lated from  the  constants  up  to  the  seventh  is  £=  72.4. 

Taking  the  first  and  third  constants  only,  gives  E  —  72.3, 
which  is  correct  to  a  close  approximation. 

The  value  of  E  found  by  plotting  the  curve  to  polar 
co-ordinates  is  E  =  72^. 

The  example  which  has  been  taken  fairly  represents  the  com- 
plexity of  the  average  distorted  waves.  Some  alternating-cur- 


0   10  20   -30  40   50  60 


80  90  100.  aiO  120  130  140  .150  JGO  170  180 

Fig-.  C 


rent  curves  are  so  greatly  distorted  that  a  larger  number  of 
terms  of  the  series  is  required  to  closely  represent  them,  but  for 
practical  purposes  three  or  four  terms  are  always  sufficient.  In 
a  large  number  of  waves  the  forms  are  so  simple  that  two  terms 
of  the  series  give  ample  approximation  for  practical  purposes. 
Examples  of  waves  of  greater  or  less  complexity  than  that 
already  operated  upon  are  given  in  figures  C,  £>,  and  E  of 


APPENDIX   A. 


701 


this  appendix  and  various  figures  of  the  text.  Figure  C  is  a 
curve  given  by  Steinmetz*  for  which  the  constants  up  to  the 
i 3th  are 


0j  =  4-  109.5 
a3  =  -  12.8 
a5  =—  22.8 
a-t  =  —  12.4 


•55 
2-95 
•595 


»  =-    3-25 
5  =  —  10.6 

=  4  7-87 
9  =  4    .245 

n  =  —    4-2 


The  ninth  and  higher  constants  are  practically  negligible,  so 
that  the  curve  may  be  represented  by  the  formula 

2=109.5  sin  a      —12.8  sin  3  a      —22.8  sin  5  a   ,  —12.4  sin  7  a 
-j-    10.5  cos  a      —3. 25  cos  3  a      —  io.6cos5a      +7. 87  cos  7  a 


0          45 


135°    180°   225°    270°   315°   360°   45° 


Fig.  D 

Figure  D  is  a  curve  given  by  Fleming  t  after  the  results  of 
tests  by  Merritt  and  Ryan.     Its  equation  is 

e  =  .196  sin  (a  —  48°  55')  +  .048  sin  (3  a  —  76°  50') 
+  .016  sin  (5  a  —  90°) 

*  Trans.  Amer.  Inst.  E.  E.,  Vol.  12,  p.  476. 

f  Fleming's  Alternate  Current  Transformer  in    Theory  and  Practice, 
Vol.  II.,  p.  454. 


702 


ALTERNATING   CURRENTS. 


=  +  .129 


a= 


=  -.047 

=  —  .016 


The  component  sinusoids  of  this  curve  are  given  in  the  figure. 
Figure  E  is  another  curve  given  by  Steinmetz,  which  is  almost 


S 

,/ 

' 

X 

\ 

^ 

/ 

\ 

x 

/ 

S 

/ 

\ 

\ 
\\ 

- 

r 

\ 

/ 

k\ 

'i 

\ 

/ 

x 

y 

\ 

t 

\ 

/,' 

\ 

I 

l^, 

I 

\ 

^T 

r-2 

)      3 

»  < 

>    i 

0     6 

»     1 

)      8 

)      <J 

0     1 

W   1 

o^ 

0  ir 

0    1 

0    1 

0   K 

0—1- 

•^ 

~n~ 

| 

E 

--T! 

Fig.  E 

a  true  sinusoid  and  may  be  represented  by  one  term.     The 
constants  are 

a  =  146.5  b=  2.6 

and  the  equation  is 

e  =  146.5  sin  (a  —  i°) . 

The  errors  of  observation  in  experimentally  determining  such 
a  curve  are  greater  than  the  deviations  of  this  curve  from  a 
sinusoid,  so  that  its  equation  may  properly  be  written 

e  =  146.5  sin  a. 

A  theoretical  discussion  of  Fourier's  series  will  be  found  in 
the  first  three  chapters  of  Byerly's  Fourier's  Series  and  Spheri- 
cal Harmonics ;  and  the  following  articles  are  of  considerable 


APPENDIX   B. 


703 


interest  in  this  connection :  Periodic  Functions  Developed  in 
Fourier  Series  ;  The  Graphical  Method,  by  Professor  John 
Perry,  London  Electrician,  Vol.  35,  p.  285  ;  Wave  Form  Syn- 
thesis, London  Electrician,  Vol.  35,  p.  257. 

APPENDIX   B. 


CHARACTERISTIC    FEATURES   OF   DIFFERENT    FORMS   OF   ALTERNAT- 
ING-CURRENT  AND    PRESSURE    CURVES. 


o 

2 

2 

a 

2 

II 

t< 

bjoj 

|| 

|1 

-s2J 

Name  of  Curve. 

H 

> 

w  s 

«  « 

&-« 

a 

'S   fl 

°F 

*o  > 

*0  S) 

*s  ^i 

g, 

•5*1 

ij 

.0  t3 

•S§ 

sH 

* 

** 

2s 

S3 

w< 

Triangle  . 

C77 

I    ICC 

•2'7'7 

9 

*5 

'Jii 

•73 

1<155 

'OJO 

Approximate  Sinusoid    . 

— 

.637 

.707 

1.414 

1.  112 

.500 

Sinusoid      

1  20 

6^7 

7O7 

I    112 

Parabolic  Curve    .... 

123 

.666 

•73° 

1.369 

1.096 

•533 

Semicircle 

0 

8 

i  198 

I  063 

fi 

•7  5 

35 

•  97 

Approximate  Rectangle  . 

122 

.856 

.889 

1.124 

1.038 

.791 

Rectangle    

121 

I  OOO 

I  OOO 

I  OOO 

I  OOO 

I  OOO 

APPENDIX   C. 

OSCILLATORY    DISCHARGES. 

The  discharge  of  a  condenser  in  a  circuit  containing  resist- 
ance is  considered  in  Section  31  and  following  sections  of  the 
text,  and  the  mutually  neutralizing  effect  of  self-inductance 
and  capacity  is  fully  explained  in  later  sections.  The  con- 


704  ALTERNATING   CURRENTS. 

ditions  brought  about  by  the  discharge  of  a  condenser  through 
an  inductive  circuit  are  not  entered  upon  in  the  text,  and  as 
they  have  some  incidental  interest  to  the  electrical  engineer 
they  will  be  explained  here. 

If  a  condenser  of  capacity  s,  charged  to  a  difference  of  poten- 
tial or  electric  pressure  E,  be  introduced  into  an  electric  circuit, 
it  will  at  once  discharge  ;  that  is,  it  will  send  a  current  through 
the  circuit  and  thus  bring  the  difference  of  potential  of  its 
plates  to  zero.  At  any  instant  the  electrical  pressure  in  the 
circuit  will  be 


where  L  and  R  are  the  self-inductance  and  resistance  of  the 
circuit.  From  the  fundamental  definition  of  a  condenser, 

q        ,  dq 

e  =  -  and  c  =  --  ±-, 
s  dt 

q  representing  the  quantity  of  electricity  in  the  condenser  at 
any  instant  during  the  discharge,  when  the  electrical  pressure 
is  e,  and  the  current  c.  Substituting  these  values  gives 

q_          d*q      dq 

s~     L  d?     Tt> 


R  dq  ,     q 


In  order  to  find  the  value  of  the  quantity  of  electricity  in  the 
condenser  at  any  instant,  and  thus  determine  the  rate  at  which 
the  condenser  discharges,  this  equation  must  be  solved  by  inte- 
gration.* The  characteristic  equation  is 


, 
L       Ls 

and  the  roots  of  this  determine  the  form  of  the  solution.     As 
this  is  a  quadratic  equation,  it  may  have  either  two  real  or  two 

*  Price's   Calculus,  Vol.  II.,  p.  458;   Forsyth's  Differential  Equations, 
p.  86, 


APPENDIX   C.  705 

imaginary  roots  depending  upon  circumstances ;  these  roots  are 


li  Rz       i 

\~2 


R 

~ 


x  =  -  —  -     -  —      — 
2  L       *  4  Z2     Ls 

It  is  evident  that  the  roots  are  real  when 


4  L"      Ls 
and  that  they  are  imaginary  when 

^<4A 

S 

In  the  first  case  the  solution  takes  the  form 
q  =  A?*  H-  B?*, 

and  c  =  -^-  =  -  Axi?*  -  Bxjf*, 

at 

where  A  and  B  are  constants  which  must  be  found  by  sub- 
stituting the  value  of  zero  for  /,  in  which  case  q  =  Q,  and 
c  =  o.  Whence, 

Q=  A  +  B,  and  Ax±  +  Bx^  =  o, 


from  which      A  = '— ,  and  B  == 

Hence 


and     ^  =  -- 


i>~| 


These  equations  show  that  q  and  r  never  fall  to  zero,  but  grad- 
ually decrease  according  to  a  logarithmic  function  as  /  increases. 

2Z 


706         ALTERNATING  CURRENTS. 

r>2 

The  time  constant  of  the  circuit  decreases  as  — -  approaches 

i  4  L~ 

—  in  value  and  is  a  minimum  of  \  sR  when  they  are  equal.* 

4.L 
When  ^?2<— -  the  roots  are  imaginary,  and  if  /  be  taken  to 

indicate  the  imaginary  unit  V—  i, 

_  R        .    f~i       3~tf~2 
Xl~      ^Z  +  i *  \Ls~     T« 


2Z 

Inserting  these  values  in  the  formulas  for  q  and  c  and  reducing 
to  trigonometrical  forms,  the  equations  become 


Ls 


Vzr 


From  these  formulas  we  see  that  when  the  roots  of  the  differ- 
ential equations  are  imaginary,  q  and  c  are  periodic  functions 
which  have  alternately  positive  and  negative  values,  so  that  the 
discharge  is  an  oscillatory  one.  In  other  words,  when  the  con- 
denser is  discharged,  during  the  first  flow  of  current  a  certain 
amount  of  energy  has  been  stored  in  the  magnetic  field  and  in 
the  return  of  this  to  the  circuit  the  condenser  is  charged  up  in 
the  opposite  direction.  This  is  repeated  over  and  over  again 
with  incredible  rapidity  but  with  decreasing  intensity,  until  the 
total  energy  of  the  original  charge  is  dissipated  in  overcoming 

the  resistance  of  the  circuit.     The  current  passes  through  one 

V~         ^T 
passes  through  all  values 

1-rS  4  J-' 

from  o  to  2  TT  and  therefore  the  period  T—  — ;  _,  and  if 


Ls      D 


*  Lodge,  On  the  Influence  of  Self-induction  on  the  Rate  of  Discharge 
of  a  Condenser,  Land.  Electrician,  Vol.  21,  p.  39. 


APPENDIX   C.  707 


R  is  very  small  compared  with  L  this  becomes  T=  2  ir 
The  period  of  oscillation  set  up  in  any  circuit  may  therefore  be 
controlled  by  increasing  Z.  By  this  means  Professor  Lodge 
succeeded  in  getting  periods  a  considerable  fraction  of  a  second 
in  length,  but  in  general  the  discharge  of  a  condenser  may  be 
said  to  be  practically  instantaneous.  If  iron  cores  are  used  in 
self-inductance  coils  for  use  with  oscillating  discharges  they 
must  be  very  finely  subdivided,  or  the  excessive  foucault 
currents  set  up  in  the  outer  layers  of  the  cores  screen  the 
inner  parts  from  any  magnetic  effects. 


Fig.  F 


The  formulas  for  the  discharge  of  a  condenser,  through  an 
inductive  circuit,  apply  equally  well  to  the  charging  current, 
which  may  be  logarithmic  or  oscillating  depending  upon  whether 

— -  ^  —     Figure  F  shows  the  dying  away  of  the  charge  and 

4  -Zy"*  ^  JLfS 

the  oscillations  of  the  discharge  current  in  an  oscillating  circuit. 
The  curve  which  touches  the  maximum  points  of  the  quantity 
curve  is  logarithmic,  and  a  similar  curve  similarly  touching 
the  current  curve  would  be  logarithmic.  Figure  G  shows  the 
growth  and  dying  away  of  a  current  due  to  a  transient  pressure 
in  an  oscillating  circuit.  Figure  H  shows  the  curve  of  dis- 
charge and  of  the  discharging  current  in  a  non-oscillating  circuit. 


ALTERNATING   CURRENTS. 


The  oscillating  electric  circuit  may  be  likened  to  a  pendulum 
or  an  oscillating  spring  (Fig.  /).  Such  a  spring  will  have  a 
period  of  vibration  dependent  upon  the  mass  (inertia)  of  its 


Fig.  G 

load,  its  elasticity,  and  the  frictional  resistance  to  its  motion. 
The  formula  giving  its  period  is  exactly  similar  to  that  for  the 
period  of  an  oscillating  discharge,  putting  mass  for  self-induc- 
tance, friction  for  resistance,  and  the  reciprocal  of  elasticity 


Fig.  H 

(compressibility)  for  capacity.  When  the  spring  stands  at  its 
neutral  point  it  is  analogous  to  the  condenser  when  discharged. 
Extending  or  compressing  the  spring  is  equivalent  to  charging 
the  condenser.  If  the  resistance  to  motion  is  small  and  the 


APPENDIX   D.  709 

extended  spring  is  released,  it  will  oscillate  through  decreasing 
distances  with  an  isochronous  period  until  the  energy  stored 
in  the  spring  by  its  extension  is  used  up  in 
overcoming  the  frictional  resistance  to  its 
motion.  If  the  resistance  to  its  motion  is 
increased,  its  period  will  be  lengthened  and 
the  number  of  oscillations  decreased.  While 
if  the  resistance  is  made  sufficiently 'great  (as 
for  instance,  if  the  spring  is  immersed  in  syrup) 
the  motion  will  be  dead  beat.  This  condi- 
tion is  analogous  to  the  electric  discharge  in  a  circuit  in  which 

*!_>_L 
4Z2     Ls 

This  subject  is  treated  at  great  length  in  Fleming's  Alternate 
Current  Transformers,  Vol.  I,  p.  364,  et  seq. ;  Bedell  and  Cre- 
hore's  Alternating  Currents,  Chaps.  7  and  8 ;  and  Gerard's 
Lemons  sur  VElectricite,  3d  ed.,  Vol.  I,  p.  253,  et  seq. 


APPENDIX   D. 

ELECTRICAL   RESONANCE. 

The  deductions  of  Chapters  III.  and  IV.  of  the  text  have 
shown  very  clearly  that  self-inductance  and  capacity  in  a  cir- 
cuit may  be  made  to  neutralize  each  other  when  a  sinusoidal 
alternating  pressure  is  applied  to  the  circuit,  and  the  self- 
inductance  and  capacity  are  constant.  In  this  case  the  self- 
inductance  and  capacity  act  in  opposition,  so  that  at  each 
instant  energy  is  being  stored  or  released  in  the  magnetic 
field  at  exactly  the  same  rate  as  energy  is  being  released  or 
stored  in  the  charge  of  the  condenser.  The  self-inductance 
and  capacity  may  therefore  be  said  to  supply  each  other's 
demands,  and  the  pressure  impressed  on  the  circuit  may  be 
wholly  utilized  in  doing  work  on  a  non-reactive  receiver,  such 
as  incandescent  lamps,  and  in  heating  the  wires  of  the  circuit. 


710  ALTERNATING   CURRENTS. 

The  actual  energy  which  is  transferred  back  and  forth  between 
the  self-inductance  and  capacity  may  be  many  times  as  great  as 
that  given  to  the  circuit  by  the  generator,  and  the  pressure  at 
the  terminals  of  the  self-inductance  and  of  the  condenser  must 
then  be  proportionally  greater  than  that  of  the  generator. 

This    condition    can    exist    only   when    2  irfL  =  —-*    or 

2  irfs 
Ls  —  — — ,  and  when   T=^  2  TT  VZj.f      From    the   condition 

2  -rrfL  —  —^—-  it  is  seen  that  -  =  2ir^/~Ls,  and  -  is   therefore 

2  T/J  /  / 

equal  to  T,  the  natural  period  of  the  circuit.      The  natural 

period  of  discharge  of  the  circuit  is  therefore  exactly  equal  to 
the  period  of  the  impressed  pressure,  or,  as  we  may  say,  to  the 
actual  rate  of  the  electrical  vibrations  impressed  on  the  circuit 
by  the  generator.  This  relation  between  the  vibrations  of  the 
line  and  of  the  generator  is  similar  to  that  of  a  vibrating  tuning 
fork  or  string  and  its  sounding  board  when  they  are  in  reso- 
nance, and  therefore  the  term  Electrical  Resonance  has,  on 
account  of  the  analogy,  been  applied  to  the  electrical  circuit. 
An  electrical  circuit  is  said  to  be  in  resonance  with  an  impressed 
pressure  when  the  natural  period  of  the  circuit  is  equal  to  the 
period  of  the  impressed  pressure.  When  this  condition  exists, 
the  maximum  current  is  caused  to  flow  in  the  circuit  by  the 
application  of  a  given  impressed  pressure,  the  value  of  the  cur- 
rent in  a  resonant  circuit  from  which  no  external  work  is  sup- 
plied being 

r*  E  E       '  ST          I 

C  =  — ,  =  — ,   since    2  TTjL  =  — 7- 

I      V          R  27T/JT 


If  the  self-inductance  and  capacity  are  in  series  in  the  circuit, 
it  is  evident  that  when  the  circuit  and  applied  pressure  are  in 
resonance  the  pressure  between  the  terminals  of  the  capacity, 

(C  \ 
},|  is  a  maximum,  since  the  circuit  current  is  a  maxi- 
2  W 

*  Text,  Chapters  III.  and  IV.     f  Appendix  C.     J  Text,  Section  33. 


APPENDIX   D.  711 

mum.  If  either  the  frequency,  the  self-inductance,  or  the 
capacity  is  changed  in  value,  the  value  of  the  current  falls, 
and  the  condenser  pressure  falls,  unless  the  other  elements  are 
changed  in  value  in  such  a  way  as  to  continue  the  condition  of 
resonance.  A  condenser  in  a  resonant  circuit  may  be  used 
as  a  transformer  of  pressure  by  connecting  non-reactive  appara- 
tus across  its  terminals,  as  has  been  suggested  by  Blakesley,* 
Loppe"  et  Bouquet,  f  Pupin,|  and  others. 

If  the  self-inductance  and  capacity  are  in  parallel  in  the  cir- 
cuit, the  pressure  at  their  terminals  cannot  be  greater  than 
that  impressed  upon  the  circuit  minus  the  loss  of  pressure  in 
the  lead  wires,  but  when  the  circuit  is  resonant,  the  circuit  cur- 
rent furnished  by  the  generator  is  at  a  maximum  which  is  equal 

Tf 

to  — ,  while  the  current  transferred  between  the  inductance 
R 

and  capacity  is  also  a  maximum  which  may  be  a  great  many 
times  as  great  as  the  maximum  value  of  the  generator  current. 
Resonant  circuits  in  the  hands  of  renowned  experimenters 
such  as  Hertz,  Lodge,  and  others  have  produced  remark- 
able results,  which  have  led  to  great  advances  in  our  knowledge 
of  electricity,  while  mathematical  analysis  of  such  circuits  has  led 
to  further  discoveries.  These  results  have  caused  some  to  ex- 
pect remarkable  effects  to  be  gained  from  the  use  of  resonant 
circuits  (or  tuned  circuits,  as  they  are  sometimes  called)  for  the 
purposes  of  the  electrical  transmission  of  power.  Circuits 
which  are  installed  for  the  transmission  of  energy  over  con- 
siderable distances  (whether  the  wires  are  overhead  or  under- 
ground) always  contain  capacity  and  self-inductance  §  dis- 
tributed along  their  length.  It  would  be  possible  in  such 
lines  to  adjust  the  capacity  and  self-inductance  so  as  to  give 
resonance,  and  the  results  to  be  gained  from  so  doing  may  be 
examined  through  analogy. 

*  Blakesley's  Alternating  Current  of  Electricity,  2d  ed.,  p.  53. 
t  Loppe  et  Bouquet's  Courants  Alternatifs  Industriels,  p.  77. 
t  Pupin,  Trans.  Amer.  Inst.  E.  £.,  Vol.  10.,  p.  382. 
§  Text,  Section  47. 


712 


ALTERNATING   CURRENTS. 


A  mechanical  analogue  of  a  resonant  circuit  is  shown  in 
Fig.  J.  This  consists  of  a  tube  fitted  with  two  plungers  and 
filled  with  a  perfectly  elastic  fluid.  The  properties  of  this 
fluid  may  be  used  to  represent  electrical  quantities  according 
to  the  analogies;  fluid  velocity — electric  current;  fluid  press- 
ure—  electric  pressure;  inertia — self-inductance;  compressi- 
bility*—  capacity;  frictional  resistance — electrical  resistance. 
Now  suppose  the  fluid  to  be  without  inertia  and  perfectly  incom- 
pressible ;  then  if  plunger  A  be  moved  toward  D,  a  uniform 


a' 


Fig.  J 

current  will  be  instantly  set  up  in  the  whole  tube,  the  velocity 
of  which  is  equal  (in  proper  units)  to  the  pressure  applied  to 
the  plunger  divided  by  the  frictional  resistance.  If  plunger  A 
is  caused  to  move  up  and  down  harmonically,  the  other  plun- 
ger will  have  an  exactly  equal  synchronous  harmonic  motion. 
This  is  exactly  analogous  to  the  state  of  an  electric  circuit 
without  inductance  or  capacity.  Figure  K  shows  diagrammati- 
cally  the  state  of  the  circuit,  where  the  distance  of  the  broken 
line  from  the  heavy  line  is  equal  to  the  current  at  each  point, 
and  the  light  line  shows  the  gradual  fall  of  pressure  between 
A  and  A\  caused  by  the  resistance,  and  the  sudden  fall  of  press- 
ure at  A',  caused  by  the  external  work  done  by  plunger  A'. 

*  Compressibility  of  a  fluid  is  the  ratio  of  compression  (change  of  vol- 
ume) to  the  pressure  producing  it,  and  electrical  capacity  is  the  ratio  of  the 
charge  (change  of  quantity)  to  the  electrical  pressure  producing  it. 


APPENDIX   D 


7*3 


If  the  fluid  be  compressible  but  have  no  inertia,  it  is  evident 
that  the  motion  of  the  plunger  at  A'  will  be  less  than  that 
at  A,  which  is  analogous  to  the  decadence  of  current  as  it 
flows  along  a  circuit  having  capacity,  due  to  the  quantity  of 


Fig.  K 

electricity  entering  into  the  static  charge.  The  movements 
of  the  plungers  are  isochronous  but  not  in  synchronism.  In 
this  case  the  motion  of  the  plunger  A  will  exert  its  maximum 
pressure  when  the  fluid  is  most  compressed,  or  at  the  end  of 


Fig.  L 

its  stroke  where  its  velocity  is  least.  Hence  the  velocity  (cur- 
rent), which  is  greatest  at  the  middle  of  the  stroke,  leads  the 
pressure  by  90°  of  phase.  The  electric  circuit  corresponding 
to  this  is  shown  in  Fig.  L. 

If  the  fluid  has  inertia  but  is  incompressible,  the  velocity 
at  A  and  A'  will  be  equal,  or  the  current  through  the  circuit 


714         ALTERNATING  CURRENTS. 

will  be  uniform,  but  the  pressure  exerted  upon  piston  A  must 
be  greatest  where  the  acceleration  is  greatest,  which  is  at  the 
beginning  of  the  stroke  where  the  velocity  is  least.  Conse- 
quently the  current  lags  behind  the  pressure  by  90°.  This 
is  analogous  to  the  electric  circuit  with  self-inductance  only. 

If  the  fluid  has  both  inertia  and  compressibility,  the  column 
of  fluid  in  the  tube  will  then  take  upon  itself  the  properties 
of  all  material  elastic  bodies,  and  will  have  a  natural  rate  of 
vibration.  This  will  be,  as  proven  in  elementary  mechanics, 
proportional  to  the  square  root  of  the  density  divided  by  the 
elasticity,  or  to  the  square  root  of  the  product  of  the  inertia 
and  compressibility. 

Hence  T=  aV ' MK  where  a  is  a  constant,  J/mass,  K  com- 
pressibility, and  T  time  of  vibration.  In  this  case  if  the  plunger 
A  (Fig./)  be  moved  with  a  sinusoidal  velocity  of  period  T9  the 
fluid  will  be  thrown  into  vibrations  which  require  one  complete 
traversal  of  the  circuit  to  make  a  wave  length.  Hence  if  there 
is  no  power  taken  from  the  circuit  there  are  nodes,  or  points  of 
no  motion,  at  a  and  a',  and  antinodes,  or  points  of  maximum 
motion,  at  the  plungers.  Since  the  direction  of  motion  in  the 
two  halves  of  a  wave  are  in  opposite  directions,  the  two  plungers 
move  in  opposite  directions  in  the  tube.  As  the  velocity  of  the 
fluid  varies  from  node  to  antinode  as  a  sinusoidal  function, 
the  loss  of  power  by  friction  is  reduced  to  one-half  the  value 
which  it  has  for  an  equal  plunger  velocity  in  the  inertialess, 
incompressible  fluid. 

Since  the  velocity  of  the  fluid  falls  off  from  plungers  to  the 
nodes,  the  pressure  upon  the  fluid  exerted  by  the  plungers  must 
be  proportionally  multiplied  at  the  nodes,  in  order  that  the 
same  power  may  be  transmitted  there  as  was  applied  at  the 
prime  plunger  A.  The  condition  of  pressure  and  velocity  is 
diagrammatically  represented  in  Fig.  M.  If  power  is  transferred 
to  an  outside  object  by  plunger  A',  it  is  impossible  for  the 
velocity  at  the  nodal  points  to  be  zero,  but  it  must  be  sufficiently 
great  to  transfer  the  power  through  the  nodal  point  with  the 


APPENDIX   D. 


715 


pressure  at  that  point.  The  relative  motions  of  the  plungers, 
under  the  conditions  here  cited,  require  that  the  power  be 
transferred  from  one  to  the  other  wholly  through  its  absorption 
and  redelivery  by  the  fluid  through  the  effects  of  inertia  and 
elasticity.  The  fluid  must  therefore  have  a  sufficient  mass  so 
that,  at  the  slow  velocity  of  the  nodes,  its  kinetic  energy  shall 
be  sufficient  to  carry  the  energy  in  the  circuit  across  the  nodal 
points. 


Fig.  M 

This  analogue  fully  represents  the  conditions  in  the  resonant 
electric  circuit.  Carrying  the  analogue,  and  the  diagrammatic 
representation  of  current  and  pressure  in  Fig.  M,  in  mind,  it  is 
easy  to  draw  definite  conclusions  in  regard  to  the  effect  of 
resonance  on  the  operation  of  circuits  for  the  transmission  of 
power  by  currents  of  electricity. 

The  advantages  of  a  resonant  circuit  for  electrical  transmis- 
sion are  then  :  (i)  a  gain  of  upwards  of  one-half  of  the  C^R 
loss  that  would  be  caused  by  the  transmission  of  an  equal 
amount  of  power  at  an  equal  receiving  pressure  over  the  same 


716         ALTERNATING  CURRENTS. 

circuit  when  out  of  resonance  ;  (2)  more  satisfactory  regulation 
than  would  be  found  in  a  non-resonant  but  reactive  line,  since 
the  difference  in  pressure  between  generator  and  receiver  is 
equal  to  current  times  resistance  instead  of  current  times  an 
impedance  which  is  greater  than  the  resistance. 

The  principal  disadvantage  of  a  resonant  circuit  for  electrical 
transmission  is  :  a  very  large  excess  of  pressure  on  the  line 
at  certain  points,  or  nodes  of  current,  which  excess  decreases 
toward  the  antinodes.  If  satisfactory  resonance  is  to  be  gained 
by  adjusting  the  self-inductance  and  capacity  of  the  circuit  so 
that  the  pressure  at  the  nodes  is  no  greater  than  ten  times  that  at 
the  antinodes,  the  average  pressure  along  the  line  must  be  caused 

to  be  seven  times  (  —  —  )  that  of  the  antinodes,  using  a  sinusoidal 


function.  In  other  words,  if  the  pressure  which  is  safe  for  use 
is  limited  by  the  insulation,  we  may  say  that  the  average  thick- 
ness of  insulation  on  the  line  must  be  seven  times  as  great  as 
would  be  necessary  at  the  generators.  This  enormous  increase 
of  insulation  must  be  made  to  save  fifty  per  cent  of  the  C^R 
loss  caused  by  the  transmission  of  a  certain  amount  of  power 
over  a  given  line.  A  much  more  reasonable  plan  would  be  to 
reduce  the  self-inductance  and  capacity  of  the  line  to  a  mini- 
mum, avoiding  resonance  and  raising  the  generator  pressure 
to  1.4  its  previous  value.  Now  the  same  power  could  be  trans- 
ferred over  the  line  with  the  same  resistance  as  before,  the  C*R 
loss  being  the  same  as  when  the  line  was  resonant,  but  the 
average  strain  on  the  insulation  would  be  only  one-fifth  as 
great  as  in  the  resonant  line. 

The  highest  pressure  which  can  be  economically  used  on  cir- 
cuits for  the  electrical  transmission  of  power  over  long  distances 
is  generally  conceded  to  be  set  at  the  limit  which  may  be 
properly  insulated.  If  this  is  true,  the  preceding  paragraph 
shows  that,  with  equal  insulation,  the  generator  pressure  may 

y 

be  safely  made  —  -  times  greater  on  3  non-resonant,  long-dis- 

V? 


APPENDIX   D.  717 

tance  transmission  line  than  that  which  is  safe  on  a  resonant 
line,  where  X  is  the  ratio  of  the  maximum  pressure  to  the 
generator  pressure  on  the  resonant  line.  This  shows  that  the 
non-resonant  line  would  be  by  far  the  most  economical  for 
long-distance  transmission  of  power,  even  if  it  were  com- 
mercially possible  to  maintain  resonance  on  service  circuits. 
For  the  distribution  of  power  over  short  distances,  the  pressure 
is  usually  quite  low,  and  the  pressure  limit  is  not  approached, 
so  that  resonance  might  be  introduced  without  adding  to  the 
insulation  ;  but  the  reactions  of  transformers  and  motors  on  the 
line  make  it  practically  impossible  to  keep  the  line  in  reso- 
nance. Similar  defects  are  seen  in  the  propositions  for  using 
resonant  lines  for  various  other  classes  of  electrical  transmission. 
These  deductions  in  regard  to  resonance  have  been  made 
upon  the  assumption  of  exactly  sinusoidal  currents.  In  practice 
these  are  now  seldom  met,  since  iron-cored  transformers  and 
motors,  and  tooth-cored  alternators,  introduce  distortions,  and 
a  circuit  which  is  resonant  for  the  fundamental  wave  is  not 
resonant  for  its  harmonics.  As  the  question  of  resonance  now 
rests,  it  does  not  present  any  opportunities  for  application  in 
practice,  nor  does  it  enter  into  problems  relating  to  ordinary 
electric  circuits  in  such  a  way  as  to  modify  practice.  In  some 
cases  of  long-distance  transmission  of  power  by  alternating 
currents  with  a  distorted  wave  of  pressure,  the  harmonics  may 
accidently  come  into  resonance  with  the  line  and  cause  an 
undue  strain  on  the  insulation ;  but  this  is  readily  guarded 
against  by  using  a  generator  which  generates  an  approximate 
sine  pressure  curve. 

Many  articles  have  been  written  upon  resonance  and  its 
effects  in  electric  circuits,  but  the  following  will  serve  to  give  a 
general  view  of  the  subject :  — 

April,  1891.  Lodge,  The  Effect  of  a  Condenser  Introduced 
into  an  Alternate-Current  Circuit,  London  Electrician, 
Vol.  26,  p.  762. 


718         ALTERNATING  CURRENTS. 

May,  1891.  Fleming,  On  Some  Effects  of  Alternating- Current 
Flow  in  Circuits  having  Capacity  and  Self-induction, 
Jour.  Inst.  E.  E.,  Vol.  20,  p.  362. 

May,  1893.  Pupin,  Practical  Aspects  of  Low  Frequency 
Electrical  Resonance,  Trans.  Amer.  Inst.  E.  E.,  Vol.  10, 

P-  370. 

June,  1894.  Anthony,  Electrical  Resonance  as  Related  to  the 
Transmission  of  Energy,  Electrical  Engineer  (N.Y.),  Vol. 

i7>  P-  545- 

October,  1894.  Blondel,  Inductance  des  Lignes  Ae"riennes 
pour  Courants  Alternatifs,  UEdairage  Electrique,  Vol.  I, 
p.  241. 

April,  1895.  Houston  and  Kennelly,  Resonance  in  Alternating- 
Current  Lines,  Trans.  Amer.  Inst.  E.  £.,  Vol.  12,  p.  133. 


INDEX. 


A. 

Active  current,  117. 

Active  pressure,  40. 

Ageing  of  transformer  cores,  539. 

All-day  efficiency  of  transformers,  492. 

Allgemeine  Elektricitats  Gesellschaft, 
induction  motors,  620,  665. 

Alternating  circuit,  current  in,  65 ; 
power  in,  109,  112;  methods  of 
measuring  power  in,  121. 

Alternating-current  curves,  charac- 
teristic features  of,  703. 

Alternating  field,  resolution  of,  647. 

Alternations,  definition,  7. 

Alternator,  7  ;  armatures,  10,  15,  16,  25, 
26,  28,  31,  35,  230,  231,  232,  234,  236, 
239.  243.  246,  366;  characteristics, 
266;  design,  239;  dimensions,  13  p 
224;  efficiency,  370;  field  excitation, 
8,  251,  268,  362;  leakage  coefficient, 
233;  losses,  221 ;  copper  losses,  223, 
226 ;  foucault  current  losses,  227 ; 
hysteresis  losses,  228  ;  rectifying  com- 
mutator, 259;  testing,  371. 

Alternators,  7  ;  as  synchronous  motors, 
571 ;  combined  output  of,  322 ;  in- 
ductor, 33  ;  in  parallel,  326  ;  in  series, 
322  ;  on  separate  feeders,  336. 

.American  transformers,  tests  of,  504. 

Amperemeter  (three)  method  of  meas- 
uring power,  127. 

Amperemeters,  alternating  current,  267, 
277. 

Ampere-turns  on  field  of  induction  mo- 
tor, 631. 

Analytical  method  of  solving  problems, 
208,  217. 


Angle  of  lag,  42,  in  ;  method  of  meas- 
uring, in. 

Apparent  energy,  116;  resistance,  71; 
watts,  116. 

Arago  disc,  596. 

Areas  of  successive  loops  of  alternating- 
current  curves,  306. 

Armature,  action  of  short-circuited  in 
rotary  field,  595 ;  calculation  of 
alternators,  239;  classification,  15; 
collectors,  31,  32;  commutated  in 
induction  motors,  624;  conductors 
of  alternator,  232  ;  conductors,  differ- 
ential action  of  alternator,  10 ;  con- 
ductors, number  of  alternator,  234; 
conductors,  number  of  alternator, 
maximum,  236;  current  and  excita- 
tion, relation  in  synchronous  motors, 
582 ;  disc,  26 ;  drum,  16 ;  insulating 
and  core  materials,  35;  pole,  29; 
poly-phase,  connections  of,  393; 
pressure,  alternator,  i,  9;  pressure 
induction  motors,  631 ;  radiating 
surface,  231,  637;  reactions  in  alter- 
nators, 246,  250;  reactions,  effect 
on  parallel  operation,  360;  reactions 
of  poly-phasers,  387 ;  resistance  in 
starting  induction  motors,  622; 
self-inductance,  243,  349,  368,  581, 
641 ;  ventilation,  230 ;  winding,  15, 
612. 

Arrangement  of  conductors  in  trans- 
former windings,  531. 

Auto-transformer,  545  ;  for  starting  in- 
duction motors,  621. 

Average  pressure  or  current,  4,  9. 

Ayrton,  testing  alternators,  377  ;  tracing 
curves,  301. 


719 


720 


INDEX. 


Ayrton  and  Perry's  inductance  stand- 
ard, 98,  420;  secohmmeter,  105,  415, 
416,  419. 

Ayrton  and  Sumpner,  transformer  test- 
ing, 487. 

B. 

Balanced  poly-phase  systems,  547. 
Ballistic  galvanometer,  57. 
Ballistic  method  of  tracing  curves,  289. 
Bedell's    contact   maker,  302;    test  of 

hedgehog  transformer,  495 ;  tracing 

curves,  300. 
Bedell  and  Ryan,  synchronous  motor 

experiments,  586. 
Blakesley,  measurement  of  power,  112, 

128,   129;    split   dynamometer,    128, 

483- 
Blondel,  contact  maker,  301 ;   tracing 

curves,  305. 
Booster,  542. 
Brown,   C.   E.    L.,   induction   motors, 

660;    parallel   operation   of  alt.rna- 

tors,  361. 

C. 

Calculation  of  core  and  windings  for 
impedance  coils,  541 ;  for  induction 
motors,  633;  for  transformers,  519. 

Calculation  of  losses,  regulation,  excit- 
ing current,  and  efficiencies  of  induc- 
tion motors,  629,  633,  640,  641 ;  of 
transformers,  461,  492,  523,  527,  529, 
530»S38. 

Calorimetric  method  of  testing  trans- 
formers, 481. 

Capacity,  78  ;  and  self-inductance  com- 
bined, 88,  89,  703,  709 ;  effect  on  reg- 
ulation of  transformers,  439 ;  press- 
ure, 80,  164 ;  required  for  Stanley 
motor,  669. 

Capacity  circuit,  178 ;  time  constant 
of,  83. 

Carey  Foster's  method  of  measuring 
mutual  induction,  409. 

Characteristics,  alternator,  266;  curve 
of  magnetization,  266;  distribution 
curve,  278;  external,  272;  loss  line, 

275- 

Checks  on  transformer  design,  531. 
Choking  coils,  541. 


Circuits  in  parallel,  57,  72,  175,  193, 
196;  in  series,  72,  159,  171,  196. 

Circumferential  velocity,  13,  230,  634. 

Coefficient  of  leakage,  233,  429,  533, 
636,  639  ;  of  mutual  induction,  398  ; 
of  self-induction,  43. 

Coils,  embedded,  24,  601,  638 ;  lathe- 
wound,  24. 

Coil  winding,  20. 

Collector  rings,  31,  32. 

Collectors,  armature,  31,  32. 

Commutated  winding,  induction  mo- 
tor, 624. 

Commutator,  rectifying,  259  ;  sparking 
of,  262;  Zipernowsky,  260. 

Compensated  voltmeter,  316. 

Compensators,  543. 

Composite  winding,  251,  319,  369. 

Composite-wound  alternators  in  par- 
allel, 369. 

Condenser,  79;  circuits,  178;  curves 
of  charge  and  discharge  of,  81 ;  en- 
ergy of  charged,  81 ;  pressure,  80, 164. 

Conducting  systems,  poly-phase,  546, 
552,  554,  555,  556;  poly-phase,  bal- 
anced, 547. 

Conductors,  arrangement  of  in  trans- 
former windings,  531. 

Constants  for  use  in  design  of  induc- 
tion motors,  634,  635,  636,  637;  of 
transformers,  463,  465,  467,  468,  532. 

Contact  makers,  301. 

Continuous  winding,  19. 

Converter,  396. 

Copper    losses,  calculation  for  trans- 

•  former,  529;  in  alternator,  223;  in 
alternator,  table,  226;  in  transfor- 
mer, 428,  436,  465 ;  in  transformer, 
effect  on  regulation,  436;  in  induc- 
tion motor,  612,  619,  640. 

Core  losses  in  alternator  armature,  227, 
228 ;  in  induction'  motor,  618,  640, 
651,  670;  in  transformer,  436,  452, 
456,  461,  523,  527  ;  in  transformer  are 
independent  of  load,  488  ;  separation 
of,  in  alternator,  372 ;  in  transformer, 
500. 

Core  materials,  38,  539,  633. 

Core  of  transformer,  ageing,  539. 

Corrections  for  wattmeter,  131. 

Cost  vs.  output  in  alternators,  383. 


INDEX. 


721 


Current,  active,  117. 

Current  and  pressure  curves,  tracing 
of,  289. 

Current  and  pressure  relations,  in  in- 
duction motors,  603  ;  in  poly-phase 
systems,  555  ;  in  synchronous  motors, 
575  ;  in  transformers,  404,  440. 

Current,  determination  of  effective,  309; 
distribution  of,  in  a  wire,  144;  ex- 
citing, of  induction  motors,  629 ;  of 
transformers,  431,  530;  in  capacity 
circuit,  85  ;  in  inductive  circuit,  65  ; 
magnetizing,  433 ;  of  induction  mo- 
tors, 601,  630 ;  of  transformers,  433, 
530 ;  relations  in  transformer,  434  ; 
rushes  in  inductive  circuit,  540 ; 
summation  zero  in  poly-phase  cir- 
cuits, 554;  wattless,  117. 

Current  density,  in  armature  conduct- 
ors, 232, 640 ;  in  collectors,  32 ;  in  field 
windings,  233,  637 ;  in  transformer 
windings,  468. 

Curve,  resolution  of,  75,  695. 

Curve  of  alternator  efficiency,  384,  385  ; 
of  motor  efficiency,  657,  658,  659, 
660,  664,  665,  670,  671 ;  of  trans- 
former efficiency,  498. 

Curve  of  magnetization,  266. 

Curve  of  pressure,  effect  of  form  on 
induction  motor,  678  ;  effect  of  form 
on  transformer,  517. 

Curve  of  squared  ordinates,  309,  312. 

Curves,  characteristic  features  of  alter- 
nating-current, 703  ;  charge  and  dis- 
charge of  condenser,  81;  of  current 
in  capacity  circuit,  81 ;  of  current  in 
inductive  circuit,  53  ;  successive  areas 
equal,  306. 


D. 

Delta  connection,  394,  551. 
Densities,   current,   33,   232,   233,  468, 
637,  640;   magnetic,  -228,  370,  462, 

635- 

Design  of  alternators,  239 ;  of  induction 
motors,  633  ;  of  transformers,  519. 

Diamond  meter,  672. 

Differential  action  in  alternator  arma- 
tures, 10 ;  in  induction  motors,  627, 
639. 

3A 


Dimensions  of  alternators,  13,  224 ;  of 
transformers,  534. 

Dimmer,  542. 

Direct  measurement  of  efficiency  of 
induction  motors,  653. 

Disc  armature,  26. 

Discharges,  oscillatory,  703. 

Distribution  of  alternating  current  in  a 
conductor,  144. 

Distribution  of  magnetism  over  alter- 
nator pole  faces,  278. 

Divided  circuits,  57,  72,  175,  193,  196. 

Double  armature  winding  for  induc- 
tion motors,  620. 

Drehfelde,  595. 

Drehstrom,  595,  601. 

Drum  armatures,  16. 

Duncan  electrodynamometer,  298. 

Duncan,  tracing  curves,  298. 

E. 

Economy  coils,  541. 

Eddy  current  losses,  alternator,  227 ; 
induction  motor,  618,  640;  trans- 
former, 432,  436,  450,  456,  461,  500, 

527- 

Effect  of  form  of  pressure  curve  on 
motor  operation,  678  ;  on  paralleling 
of  alternators,  360 ;  on  transformer 
efficiency,  517. 

Effective  pressure,  4,  8,  699 ;  and  cur- 
rent, determination  of,  from  curves, 

309. 

Efficiency  curves,  of  alternator,  384, 
385;  of  transformer,  498. 

Efficiency,  of  alternators,  370 ;  of  alter- 
nators, variation  with  output,  383; 
of  induction  motors,  652;  of  trans- 
formers, 461 ;  of  transformers,  all 
day,  492 ;  plant,  495  ;  point  of  maxi- 
mum, 500, 641 ;  weight,  of  alternators, 
383 ;  weight,  of  induction  motors, 
661 ;  weight,  of  transformers,  500. 

Electrical  resonance,  709. 

Electricity,  transfer  by  mutual  induc- 
tion, 403. 

Electrodynamometer,  Duncan,  298. 

Electrodynamometers,  267,  277. 

Electromagnetic  repulsion,  643. 

Electrometer,  method  of  measuring 
power,  121. 


INDEX. 


Electrometer,    quadrant,    121 ;     Ryan, 

294. 

Electrostatic  wattmeter,  124. 
Elwell-Parker   alternators   in  parallel, 

345- 

Embedded  windings,  24,  601,  609,  638. 

Emerson  synchronous  motor,  666. 

Emmett,  tables,  impedances  and  re- 
actances of  circuits,  144 ;  induction 
factors  and  power  factors,  120 ;  skin 
effect,  148. 

Energy,  apparent,  116. 

Energy  of  charged  condenser,  81 ;  of 
mutual  induction,  400;  of  self-in- 
duced magnetic  field,  52. 

Equalizing  connection  for  composite 
alternators  in  parallel,  369. 

Equivalent  sinusoid,  78. 

Ewing's  experiment  on  iron  losses,  488. 

Exciter,  8,  313,  318. 

Exciting  current,  431 ;  of  induction 
motor,  629 ;  of  transformer,  530. 

External  characteristic,  alternator,  272. 

Extra  current,  135. 


Factor,  impedance,  143 ;  induction,  120 ; 

power,  116. 
Features  of  alternating-current  curves, 

703- 

Ferranti  alternator,  345  ;  meter,  672 ; 
transformer,  512. 

Field,  resolution  of  alternating,  647 ; 
rotary,  591  ;  rotary,  constant,  597  ; 
rotary,  definition,  600;  rotary,  irregu- 
lar, 593 ;  strength  in  synchronous 
motors,  573 ;  strength  in  synchron- 
ous motors,  maximum  power  factor, 
580 ;  strength  in  relation  to  armature 
current  in  synchronous  motors,  582 ; 
turns  per  volt  in  induction  motors, 
635  ;  windings,  alternators,  251  ; 
windings,  induction  motors,  614,  634, 
650 ;  windings,  induction  motors,  dif- 
ferential action,  627. 

Field  ampere-turns,  induction  motors, 
631. 

Field  current,  wavy  alternator,  268. 

Field  excitation,  alternator,  8,  251,  268, 
362;  composite,  251,  319,  369;  sep- 
arate, 251, 313, 321 ;  self,  251,  320,  321 , 


Field  frequency,  601. 

Field  induced  pressure,  induction 
motor,  625. 

Field  resistance,  starting  induction 
motors,  620. 

Fleming,  tracing  curves,  305;  trans- 
former tests,  485,  488,  495,  511. 

Ford,  transformer  tests,  504. 

Form  of  pressure  curve,  influence  of, 
360,  517,  678,  717. 

Fort  Wayne  synchronous  motor,  666. 

Foucault  current  losses,  alternator,  227  ; 
induction  motor,  618,  641 ;  trans- 
former, 432,  436,  450,  456,  461,  500, 
527 ;  calculation  for  transformer,  527  ; 
effect  on  transformer  current,  456. 

Fourier's  series  applied  to  alternating 
curves,  75,  695. 

Frequency,  6  ;  alternator,  7,  8,  227,  341, 
665 ;  effect  of,  on  induction  motors, 
663  ;  effect  of,  on  parallel  operation, 
341 ;  effect  of,  on  transformers,  513. 

G. 

Galvanometer,  shunted  ballistic,  57. 

Ganz  &  Co.,  regulator,  313. 

General  Electric  Co.,  alternators  in 
parallel,  346 ;  induction  motors,  622- 
624,  665  ;  monocyclic  system,  673  ; 
regulators,  318. 

Gerard,  tracing  curves,  290. 

Gordon,  alternators  in  parallel,  345 ; 
on  parallel  operation,  345. 

Graphical  construction  of  pressure 
curve,  284,  285. 

Graphical  determination  of  induction 
motor  relations,  603  ;  of  synchronous 
motor  relations,  575  ;  of  transformer 
relations,  440. 

Graphical  solutions  of  problems,  151 ; 
in  parallel  circuits,  175;  in  parallel 
circuits,  conclusions,  193;  in  series 
circuits,  159 ;  in  series  circuits,  con- 
clusions, 171;  in  series  and  parallel 
circuits  combined,  196. 


Hanson  and  Webster,  experiments  on 

rotary  field,  597. 
Hedgehog  transformer,  495,  512. 


INDEX. 


723 


Henry,  44,  47. 

Hopkinson,  testing  transformers,  477 ; 
on  parallel  operation,  341. 

Hysteresis  loss  in  alternators,  228 ;  in 
induction  motors,  618,640;  in  trans- 
formers, 432,  436,  450,  455.  461,  500, 
523 ;  in  transformers,  calculation  of, 
523;  in  transformers,  effect  on  excit- 
ing current  452. 

I. 

Ideal  induction  motor,  607. 

Ideal  transformer,  regulation,  437. 

Idle  current,  118. 

Impedance,   definition,   71,  72;    coils, 

541;  factor,  143. 
Impedance  coils  in  transformer  circuit, 

43i- 

Impressed  pressure,  40. 

Impulsive  current  rushes,  540. 

Inductance,  42,  397. 

Inductance  standards,  98,  420. 

Induction  densities  (see  Magnetic  den- 
sities). 

Induction  factor,  120. 

Induction  motor,  armature  winding, 
612,  638 ;  design,  633 ;  differential 
action  in  fields,  627,  639;  effect  of 
form  of  pressure  curve,  678  ;  effect  of 
frequency,  663  ;  efficiency,  652  ;  excit- 
ing current,  629 ;  field  ampere-turns, 
631 ;  field  windings,  614,  625,  634, 
650 ;  formula  from  transformer,  628  ; 
leakage,  607,  609,  633,  636,  639,  641 ; 
maximum  load,  612;  monocyclic 
system,  676;  output  and  pressure, 
642 ;  power  factor  measurement,  654  ; 
regulation,  655 ;  rotary  field,  591 ; 
single-phase,  644 ;  single-phase,  form- 
ula for,  650 ;  slip,  601, 631 ;  speed,  601 ; 
Stanley,  667  ;  starting  and  regulating 
devices,  619;  torque  and  regulation, 
655;  torque  diagram,  611;  wattless 
current,  630;  weight  efficiency,  661. 

Induction  motor  armature,  definition 
of,  600. 

Induction  motor  field,  definition  of, 
600. 

Induction  motors,  miscellaneous,  666. 

Inductive  circuit,  definition,  178  ;  effect 
of  breaking,  56,  135. 


Inductive  resistance,  71. 

Inductor  alternator,  33. 

Influence  of  form  of  pressure  curve, 
360,  517,  678,  717. 

Instruments,  277,  382. 

Insulating  materials,  35. 

International  Electrical  Congress,  the 
henry,  44  47. 

Iron  losses,  constant,  Ewing's  experi- 
ment, 488 ;  in  alternators,  227,  228 ; 
in  induction  motors,  618,  640,  651, 
670 ;  in  transformers,  436,  452,  456, 
461,  500,  523,  527;  in  transformers, 
effect  on  regulation,  436;  in  trans- 
formers, relation  to  load,  488. 

Irregular  rotary  field,  593. 

J. 

Joints  in  magnetic  circuit  of  trans- 
former, 538. 

Joubert,  tracing  curves,  28,  291. 
Joubert's  contact  maker,  301. 

K. 

Kapp  alternators  in  parallel,  345 ;  trans- 
former, 512. 

Kennelly,  distribution  of  current  in  a 
wire,  149 ;  impedance  factor,  143. 

Kolben,  design  of  induction  motors, 
635  ;  magnetic  densities,  371. 


Lag  angle,  42;  measurement  of,  in. 
Lamp  with  impedance  coil,  541. 
Lap  or  loop  winding,  20. 
Leakage,  coefficient,  233,  429,  533,  607, 

636;  in  alternators,  233  ;  in  induction 

motors,  607,  609,  633,  636,  639,  641 ; 

in  transformers,  429  ;  in  transformers, 

calculation  of,  533. 
Leakage  current,  432. 
Load,  synchronous  motor  maximum, 

583- 

Loop  or  lap  winding,  20. 

Loss  line,  alternator,  275. 

Losses  in  alternators,  221 ;  in  alterna- 
tors, copper,  223,  226 ;  in  alternators, 
foucault  current,  227  ;  in  alternators, 
hysteresis,  228  ;  in  induction  motors, 
copper,  612,  619,  633,  640;  in  induc- 
tion motors,  foucault  current,  618, 


724 


INDEX. 


640 ;  in  induction  motors,  hysteresis, 
618,  640 ;  in  transformers,  copper, 
436,  465,  529;  in  transformers,  fou- 
cault  current,  432,  436,  450,  456,  461, 
500,  527 ;  in  transformers,  hysteresis, 
432,  436,  450,  455,  461,  500,  523. 

M. 

Magnetic  circuit,  joints  in  transformer 
538. 

Magnetic  densities  in  alternator  arma- 
tures, 228,  370 ;  in  induction  motors, 
635  ;  in  transformers,  462. 

Magnetic  distribution  curve,  278. 

Magnetic  field,  energy  of  self-induced, 
52 ;  rotary,  591. 

Magnetic  leakage,  in  alternators,  233 ; 
in  induction  motors,  607,  609,  633, 
636,  639,  641 ;  in  transformers,  429 ; 
in  transformer,  calculation  of,  533. 

Magnetic  reluctance,  effect  on  trans- 
former, 450,  492. 

Magnetic  shunt  transformer,  448. 

Magnetization  curve,  alternator,  266. 

Magnetization  of  transformer  core,  433. 

Magnetizing  current,  433;  induction 
motor,  601, 630 ;  transformer,  433,530. 

Magneto  machines,  25. 

Maximum  efficiency,  point  of,  in  induc- 
tion motors,  641 ;  in  transformers, 
500. 

Maximum  load  of  induction  motors, 
612 ;  of  synchronous  motors,  583. 

Maximum  output  of  alternators,  274. 

Maxwell,  measurement  of  mutual  in- 
ductance, 412,  416 ;  measurement  of 
self-inductance,  96. 

Maxwell  and  Rayleigh,  measurement 
of  self-inductance,  93. 

Measurement  of  lag  angle,  in ;  of 
mutual  inductance,  406 ;  of  power  in 
single-phase  circuits,  121 ;  of  power 
in  poly-phase  circuits,  556;  of  self- 
inductance,  90. 

Merritt  and  Ryan,  transformer  tests, 
470. 

Mershon,  tracing  curves,  296. 

Mesh  connection,  394,  551. 

Meter,  Ferranti's,  672 ;  Scheeffer's,  672 ; 
Shallenberger's,  670 ;  Thomson's, 
673. 


Monocyclic  system,  673. 

Mordey,  on  parallel  operation ,339, 343, 
358;  on  transformer  frequency,  515; 
testing  alternators,  375  ;  testing  trans- 
formers, 480. 

Motor-generator  method  of  testing  al- 
ternators, 379. 

Multi-phase,  387. 

Multi-phaser,  387. 

Mutual  inductance,  397,  398;  meas- 
urement, 406;  measurement  by 
amperemeter  and  voltmeter,  407; 
measurement  by  amperemeter  and 
galvanometer,  408 ;  measurement  by 
comparison,  416 ;  comparison  with 
capacity,  409,  411 ;  comparison  with 
self-inductance,  412,  415;  secohm- 
meter  in  measuring,  415,  416,  419. 

Mutual  induction,  396;  coefficient  of, 
398;  coils  with  iron  cores,  419; 
energy  of,  400 ;  of  distributing  cir- 
cuits, 421 ;  of  poly-phase  circuits, 
556 ;  transfer  of  electricity  by,  403. 


Neutral  point,  394. 

Niven,  measurement  of  mutual  induc- 
tance, 415. 

0. 

Oerlikon  induction  motors,  658,  665. 

Open-circuit  current,  433. 

Oscillatory  discharges,  703. 

Output  proportional  to  square  of  press- 
ure in  induction  motors,  642. 

Output  vs.  efficiency,  weight,  and  cost, 
alternators,  383;  induction  motors, 
661 ;  transformers,  500. 

P. 

Parallel,  alternators  in,  326. 

Parallel  circuits,  57,  72,  175,  193,  196; 
mutual  inductance  of,  421 ;  pressure 
and  current  relations,  72,  193. 

Parallel  distributing  circuits,  mutual 
inductance  of,  421. 

Parallel  operation,  326;  conclusions, 
367;  effect  of  armature  reactions, 
360 ;  effect  of  form  of  pressure  curve, 
360 ;  effect  of  frequency,  341 ;  effect  of 
self-inductance,  349 ;  EKvell-Parker 


INDEX. 


725 


alternators  in,  345 ;  Ferranti  alter- 
nators in,  345;  Ganz  alternators  in, 
345 ;  General  Electric  alternators  in, 
346;  Gordon  alternators  in,  345; 
Gordon  on,  345 ;  Hopkinson  on, 
341 ;  Kapp  alternators  in,  345 ; 
Mordey  on,  339,  343,  358  ;  regulation 
for,  362 ;  Stanley  alternators  in,  345  ; 
Steinmetz  on,  346,  349,  358  ;  success 
°f|  338 ;  Thomson-Houston  alter- 
nators in,  341 ;  usual  practice,  335  ; 
wattless  current,  353  ;  Westinghouse 
alternators  in,  341. 

Parallel  wires,  self-inductance  of,  140. 

Period  of  alternating  current,  6. 

Periodicity,  7. 

Periphery  velocity,  13,  230,  634. 

Permeability,  effect  of  variable,  107 ; 
419, 450 ;  on  mutual  inductance,  399 ; 
on  self-inductance,  45. 

Phase,  40. 

Phase  diagram,  definition,  157. 

Phase  indicator,  330. 

Phase-splitter,  572,  652,  688. 

Phase  transformation,  686. 

Phasing  current,  347. 

Pirani,  measurement  of  mutual  induc- 
tance, 411;  of  self-inductance,  103. 

Pitch  of  poles,  12. 

Plant  efficiency,  495. 

Polar  curves  of  current  and  pressure, 
309,  700. 

Pole  armatures,  29. 

Polygon  of  currents  or  pressures,  42, 
73,  87,  151. 

Poly-phase,  387. 

Poly-phaser,  387. 

Poly-phase  circuits,  connecting  up,  393, 
546,  548, 683 ;  self  and  mutual  induc- 
tion of,  556. 

Poly-phase  conducting  systems,  546; 
connecting  up  armatures,  393 ;  ma- 
chines, 387;  systems,  balanced,  547; 
systems,  current  summation  zero, 
554;  systems,  power  uniform,  552; 
relations  of  currents  and  pressures 
in,  555- 

Poly-phase  transformers,  683. 

Power  factor,  116. 

Power  in  alternating  circuit,  109,  112, 
measurement,  121 ;  measurement  by 


electrometer,  121 ;  by  electrostatic 
wattmeter,  124;  by  split  dynamom- 
eter, 128 ;  by  three  amperemeters, 
127 ;  by  three  instruments,  128 ;  by 
three  voltmeters,  125 ;  by  wattmeter, 

131- 

Power  in  any  poly-phase  circuit,  566. 

Power  in  poly-phase  system,  measure- 
ment, 556;  two-phase,  common  re- 
turn, 559 ;  two-phase,  independent 
circuits,  557 ;  three-phase,  one  watt- 
meter, 564;  three-phase,  two  watt- 
meters, 563 ;  three-phase,  three 
wattmeters,  560. 

Predetermination  of  losses,  regulation, 
exciting  current,  and  efficiencies  of 
induction  motors,  629,  633,  640,  641 ; 
of  transformers,  461,  492,  523,  527, 

S29,  530,  53i»  533,  538. 

Pressure,  active,  40 ;  effective,  4,  8 ; 
determination  of  effective  value  from 
curve,  309;  formula  for  induction 
motors,  625,  628,  650;  formula  for 
transformers,  427,  513 ;  formula  for 
alternators,  i,  9 ;  impressed,  40 ;  of 
alternators,  form  of  curve,  i ;  of  self- 
induction,  39 ;  triangles,  42,  73,  87. 

Pressure  and  current  relations,  induc- 
tion motors,  603 ;  poly-phase  sys- 
tems, 555  ;  synchronous  motors,  575  ; 
transformers,  404,  440. 

Pressure  curve,  effect  of  form  on  in- 
duction motor,  678 ;  effect  of  form 
on  transformer,  517  ;  graphical  con- 
struction, 284,  285  ;  tracing  of,  289. 

Prevention  of  spark  on  breaking  cir- 
cuit, 136. 

Primary  coil,  404. 

Primary  current  wave,  form  of,  450. 

Q. 

Quadrant,  44. 

Quadrant  electrometer,  121. 


Radiating  surface,  armatures,  231,  640  ; 

fields,   232,  637;   transformers,  468, 

509. 
Rate  of  work  in  inductive  circuit  when 

current  is  rising  or  falling,  60. 
Rated  motor,  testing  alternators,  375. 


726 


INDEX. 


Ratio  of  transformation,  426,  428,  466. 

Reactance,  71,  72;  coils,  541. 

Reactions,  alternator  armature,  246, 
250,  360;  induction  motors,  607,  636; 
poly-phasers,  387. 

Reactive  circuit,  definition,  178. 

Reactive  circuits  in  parallel,  57, 72, 175, 
193,  196 ;  in  series,  72,  159,  171,  196. 

Rectifying  commutator,  259. 

Regulating  devices,  alternators,  313, 
321 ;  induction  motors,  619. 

Regulating  for  constant  current,  321; 
pressure,  313. 

Regulation  by  series  exciter,  318 ;  of 
alternator,  313,  320,  321 ;  of  induc- 
tion motor,  602,  619,  632,  639,  655 ; 
of  ideal  transformer,  437 ;  of  trans- 
former, effect  of  losses,  436 ;  of  trans- 
former, effect  of  self-inductance  or 
capacity,  439;  tests  of  transformers, 
490. 

Regulator,  Ganzand  Co. ,313;  General 
Electric  Co.,  318 ;  Westinghouse, 

317. 
Relations  of  currents  and  pressures  in 

poly-phase  systems,  555. 
Reluctance,    magnetic,  of  transformer 

core,  450,  492. 

Repulsion,  electromagnetic,  643. 
Resistance,  apparent,  71. 
Resistance  triangles,  42,  73,  87. 
Resolution   of   alternating  field,   647; 

of  curves,  75,  695. 
Resonance,  709. 
Resonance  analysis,  300. 
Reversing  poly-phase  motors,  682. 
Ring  armatures,  25. 
Rise  of  temperature,  231,  466. 
Rotary  field,  591 ;  constancy,  597  ;  effect 

on    armature,   595;    effect   on   field 

windings,  625  ;  induction  motor,  591 ; 

irregular,  593. 
Rotary  transformers,  688. 
Rotating  magnetic  field,  591 ;  irregular, 

593- 
Rush  of  current  on  closing  inductive 

circuit,  540. 
Ryan  and  Bedell,  synchronous  motor 

experiment,  586. 
Ryan  and  Merritt,  testing  transformers, 

470. 


Ryan  contact  maker,  301,  302;  elec- 
trometer, 294 ;  experiment  on  trans- 
former leakage,  430 ;  tracing  curves, 
293 ;  transformer  curves,  460,  472. 

S. 

Scheeffer's  meter,  672. 

Searing  and  Hoffman  contact  maker, 
301. 

Secohm,  44. 

Secohmmeter,  105. 

Secondary  coil,  404. 

Secondary  generator,  396. 

Self-excited  alternator,  regulation,  320, 
321. 

Self-induced  field,  energy  of,  52. 

Self-inductance,  42,  43;  and  capacity 
combined,  88,  89,703,  709;  divided 
circuits,  57,  72 ;  effect  on  parallel 
operation,  349  ;  effect  on  transformer 
regulation,  439,  491 ;  measurement, 
90;  measurement  by  bridge,  93; 
measurement,  comparison  with  ca- 
pacity, 100,  103  ;  measurement,  com- 
parison with  resistance,  90 ;  measure- 
ment, comparison  with  standard 
self-inductance,  98 ;  measurement, 
comparison  of  two  self-inductances, 
96 ;  of  alternator  armature,  243,  349 ; 
of  induction  motor  armature,  607, 
622 ;  of  parallel  wires,  140. 

Self-induction,  39;  coefficient  of,  43; 
pressure  of,  39. 

Self-inductive  circuits,  current  in,  65; 
current  rushes  in,  540;  in  parallel 
and  series,  72 ;  poly-phase,  556 ; 
power  in,  109,  112;  rate  of  work  in, 
60 ;  rise  and  fall  of  current  in,  53 ; 
time  constant,  62 ;  under  sinusoidal 
pressure,  65  ;  variable  permeability, 

45,  i°7- 

Separately  excited  alternator,  regula- 
tion, 313,  321. 

Separation  of  foucault  current  and 
hysteresis  losses,  in  alternators,  372 ; 
in  transformers,  500. 

Series,  alternators  in,  322. 

Series  circuits,  reactive,  pressure  and 
current  relations  in,  72,  159,  171,  196. 

Series  and  parallel  circuits  combined, 
196. 


INDEX. 


727 


Shop  tests  of  alternators,  382 ;  of  trans- 
formers, 486. 

Siemens  and  Halske,  induction  mo- 
tors, 622,  623, 625 ;  tracing  curves,  365. 

Single-phase,  387. 

Single-phaser,  387. 

Single-phase  induction  motors,  644, 
650. 

Skin  effect,  149. 

Slip  of  induction  motor,  601,  631,  655. 

Snell,  magnetic  densities,  370. 

Sparking,  commutator,  262. 

Sparking  on  breaking    a   circuit,  56, 

135- 

Speed  of  alternators,  13,  239;  of  induc- 
tion motors,  601,  615,  634. 

Split  dynamometer,  128,  483. 

Split-phase  motor,  572. 

Standard  inductance,  98,  420. 

Standing  torque,  656. 

Stanley  alternators  in  parallel,  345 ; 
arc-light  alternator,  269,  321 ;  induc- 
tion motors,  622,  624,  667 ;  trans- 
formers, 475,  494,  498,  504. 

Star  connection,  394,  550. 

Starting  induction  motors,  619,  652. 

Starting  torque,  607,  656. 

Steinmetz,  monocyclic  system,  673 ;  on 
parallel  operation,  346,  349,  358. 

Step,  322. 

Stray  power  method  of  testing  alterna- 
tors, 375;  induction  motors,  653; 
transformers,  485. 

Surface  velocity,  13,  230,  634. 

Swinburne  electrostatic  watt-meter, 
125 ;  hedgehog  transformer,  495, 

Sis- 
Synchronism,  322. 
Synchronizers,  329. 
Synchronizing  current,  340,  349,  353. 
Synchronous  motor  method  of  testing 

alternators,  380. 

Synchronous  motors,  571 ;  field 
strength,  573,  578,  580,  582;  maxi- 
mum load,  583  ;  pressure  and  current 
relations,  575 ;  relation  of  armature 
current  to  excitation,  582. 

T. 

Table,  alternator  copper  loss,  226; 
alternator  dimensions,  14,  224 ;  alter- 


nator frequencies  and  inductions, 
371 ;  characteristic  features  of  alter- 
nating curves,  703  ;  dynamo  dimen- 
sions, 14;  efficiency,  power  factor, 
etc.  of  induction  motors,  656;  Flem- 
ing's transformer  tests,  512 ;  Ford's 
transformer  tests,  505-508,  510;  in- 
creased resistance  to  alternating  cur- 
rents, 147,  148,  149;  inductions  for 
induction  motors,  635;  maximum 
wire  diameters  for  alternating  cur- 
rents, 149;  number  of  poles  for  in- 
duction motors,  615 ;  polarized  re- 
lays, inductance  of,  49;  power  and 
induction  factors,  120;  resistance, 
reactance,  and  impedance  of  cir- 
cuits, 145 ;  self-inductance  of  alter- 
nator, 52 ;  synchronous  motor  over- 
loads, 585 ;  tests  of  transformers, 
473,  505,  506,  5°7,  5o8,  51°,  512; 
transformer  hysteresis  loss,  518 ; 
transformer  dimensions,  534 ;  weight 
of  American  induction  motors,  662 ; 
weight  of  European  induction  mo- 
tors, 662. 

Teeth,  armature,  24 ;  field,  638. 

Temperature  of  alternators,  231 ;  of 
transformers,  466. 

Tesla  motor,  591,  658. 

Testing  alternators,  371 ;  Ayrton's 
method,  377  ;  by  dynamometer,  372 ; 
Hopkinson's  method,  373 ;  Mor- 
dey's  method,  375  ;  motor-generator 
method,  379 ;  rated  motor  method, 
375 ;  shop  tests,  382 ;  synchronous 
motor  method,  380. 

Testing  induction  motors,  652;  for 
efficiency,  653 ;  for  power  factor, 
654 ;  for  regulation,  655  ;  for  torque, 

655. 

Testing  transformers,  469  ;  Ayrton  and 
Sumpner's  method,  487  ;  calorimetric 
method,  481 ;  for  regulation,  490; 
Hopkinson's  method,  477;  Mor- 
dey's  method,  480;  Ryan  and  Mer- 
ritt's  method,  470;  split  dynamom- 
eter method,  483 ;  stray  power 
methods,  485 ;  voltmeter  and  am- 
meter methods,  484;  wattmeter 
method,  484. 

Tests  of  American  transformers,  504. 


728 


INDEX. 


Thompson  on  weights  of  induction 
motors,  662. 

Thomson's  impedance  coil,  543  ;  mag- 
netic shunt  transformer,  448  ;  motor, 
645  ;  recording  wattmeter,  673. 

Three  instrument  methods  of  measur- 
ing power,  125,  127,  128. 

Three-phase,  387,  546. 

Three-phaser,  387,  546. 

Three-phase  systems,  548 ;  mesh  con- 
nection, 394,  551 ;  power  measure- 
ment, 560;  star  connection,  394, 
550;  transformers,  683,  685. 

Time  constant,  63;  examples,  64;  of 
capacity  circuit,  83  ;  of  inductive  cir- 
cuit, 62. 

Torque  diagram,  induction  motor, 
611;  of  ideal  induction  motor,  607. 

Torque  of  induction  motors,  measure- 
ment of,  655;  standing,  656;  start- 
ing, 656. 

Torque  of  synchronous  motors,  580, 
590,  690. 

Tracing  curves,  289 ;  Ayrton's  method, 
301 ;  ballistic  method,  289 ;  Bedell's 
method,  300;  Duncan's  method,  298  ; 
Gerard's  method,  290 ;  Joubert's 
method,  291 ;  Mershon's  method, 
296 ;  Pupin's  method,  300 ;  Ryan's 
method,  293. 

Transfer  of  electricity  in  mutually  in- 
ductive circuits,  403  ;  in  self-inductive 
circuits,  55. 

Transformation  of  constant  pressure  to 
constant  current,  446. 

Transformation  of  phases,  686. 

Transformer,  396 ;  ageing  of  core,  539  ; 
all-day  efficiency,  492 ;  arrangement 
of  conductors,  531 ;  calculations,  519  ; 
calculation  of  copper  and  core,  519 ; 
calculation  of  copper  losses,  529 ; 
calculation  of  exciting  current,  530; 
calculation  of  foucault  current  loss, 
527;  calculation  of  hysteresis  loss, 
523;  calculation  of  magnetic  leakage, 
533;  checks,  531;  core  losses,  461, 
523,  527;  core  losses,  separation 
of,  500;  core  magnetization,  433; 
copper  loss,  465,  529  ;  current  density, 
468  ;  current  relations,  434 ;  curves  by 
Ryan,  460 ;  efficiency,  461 ;  exciting 


current,  431 ;  for  poly-phase  circuits, 
683 ;  formula  applied  to  induction 
motor, '628;  frequency,  513;  func- 
tions of,  396 ;  hedgehog,  495 ;  mag- 
netic densities,  462 ;  magnetic  leak- 
age, 429,  533;  magnetic  leakage, 
Ryan's  experiments,  430;  magnet- 
izing current,  433;  pressure  rela- 
tions, 404,  440 ;  primary  coil,  404 ; 
radiating  surface,  468,  509;  ratio  of 
transformation,  426,  466;  regulation, 
436,  437,  439;  regulation  tests,  490; 
relations  determined  graphically, 
440 ;  rotary,  688 ;  secondary  coil, 
404;  windings,  465,  468,  513,  519, 
529,  531,  532. 

Triangle  connection,  394,  551. 

Tri-phase,  387,  546. 

Tri-phaser,'387,  546. 

Two-phase  systems,  common  return, 
548  ;  systems,  independent,  546 ;  sys- 
tems, power  measurement,  557 ;  trans- 
formers, 684. 

Two-phase,  387,  546. 

Two-phaser,  387,  546. 

U. 

Undulatory  or  wave  winding,  18. 
Uniform  rotary  field,  597. 

V. 

Variation  in  alternator  exciting  cur- 
rent, 268. 

Vector  analysis  applied  to  problems, 
208. 

Vector  diagram,  definition,  157. 

Velocity,  surface,  13,  230,  634. 

Ventilation,  armature,  230. 

Voltmeter  and  amperemeter  method  of 
testing  transformers,  484. 

Voltmeter  (three)  method  of  measuring 
power,  125  ;  Westinghouse,  compen- 
sated, 316. 

Voltmeters,  alternating,  267,  277. 

W. 

Wattless  current,  117;  effect  on  torque 
of  synchronous  motors,  590 ;  in  paral- 
lel operation,  353. 

Wattless   magnetizing   current,  indue- 


INDEX. 


729 


tion   motor,  601,   630;   transformer, 

433- 

Wattmeter,  in,  131 ;  corrections,  131; 
electrostatic,  124;  for  high  pressure, 
382;  for  measuring  power,  131; 
for  poly-phase  measurement,  556; 
method  of  testing  transformers,  484 ; 
Scheeffer's  recording,  672;  Thom- 
son's recording,  673. 

Watts,  apparent,  116. 

Wave  winding,  18. 

Wavy  field  current  in  alternator,  268. 

Webster  and  Hanson  experiments  on 
rotary  field,  597. 

Weight  efficiency,  alternators,  383 ;  in- 
duction motors,  661 ;  transformers, 
500. 

Weights  of  induction  motors,  662;  of 
transformers,  500. 


Westinghouse  compensated  voltmeter, 
316;  induction  motor,  591,622,625, 
658,665;  regulator, 317  ;  transformer, 

479,  494.  5°4.  512- 

Winding,  alternator  armature,  15 ;  al- 
ternator field,  251 ;  induction  motor 
armature,  612,  638 ;  induction  mo- 
tor field,  614,  625,  634,  650;  trans- 
former, 465,  468,  513,  519,  529,  531, 
532. 

Working  current,  117. 

Y. 

Y  connection,  394,  550. 

Z. 

Zipernowsky  commutator,  260. 


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tricity and  Magnetism.  By  H.  W.  WATSON,  D.Sc.,  F.R.S., 
and  S.  H.  BURBURY,  M.A. 

Vol.   I.    ELECTROSTATICS.    8vo.    $2.75. 

Vol.  II.    MAGNETISM  AND  ELECTRODYNAMICS.    8vo.    $2.60. 


THE    MACMILLAN    COMPANY, 

66   FIFTH   AVENUE,  NEW   YORK. 


UNIVERSITY 


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